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Properties

Label 1-1259-1259.1028-r0-0-0
Degree $1$
Conductor $1259$
Sign $-0.722 + 0.690i$
Analytic cond. $5.84677$
Root an. cond. $5.84677$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.993 + 0.114i)2-s + (0.971 − 0.237i)3-s + (0.973 − 0.227i)4-s + (−0.768 − 0.639i)5-s + (−0.937 + 0.347i)6-s + (−0.999 − 0.0149i)7-s + (−0.941 + 0.337i)8-s + (0.887 − 0.461i)9-s + (0.836 + 0.547i)10-s + (−0.962 + 0.271i)11-s + (0.891 − 0.452i)12-s + (0.858 + 0.513i)13-s + (0.995 − 0.0997i)14-s + (−0.898 − 0.439i)15-s + (0.896 − 0.443i)16-s + (−0.0374 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.993 + 0.114i)2-s + (0.971 − 0.237i)3-s + (0.973 − 0.227i)4-s + (−0.768 − 0.639i)5-s + (−0.937 + 0.347i)6-s + (−0.999 − 0.0149i)7-s + (−0.941 + 0.337i)8-s + (0.887 − 0.461i)9-s + (0.836 + 0.547i)10-s + (−0.962 + 0.271i)11-s + (0.891 − 0.452i)12-s + (0.858 + 0.513i)13-s + (0.995 − 0.0997i)14-s + (−0.898 − 0.439i)15-s + (0.896 − 0.443i)16-s + (−0.0374 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.722 + 0.690i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1028, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ -0.722 + 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02148394482 + 0.05357739697i\)
\(L(\frac12)\) \(\approx\) \(0.02148394482 + 0.05357739697i\)
\(L(1)\) \(\approx\) \(0.5976548877 - 0.09424423632i\)
\(L(1)\) \(\approx\) \(0.5976548877 - 0.09424423632i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.993 + 0.114i)T \)
3 \( 1 + (0.971 - 0.237i)T \)
5 \( 1 + (-0.768 - 0.639i)T \)
7 \( 1 + (-0.999 - 0.0149i)T \)
11 \( 1 + (-0.962 + 0.271i)T \)
13 \( 1 + (0.858 + 0.513i)T \)
17 \( 1 + (-0.0374 - 0.999i)T \)
19 \( 1 + (-0.244 - 0.969i)T \)
23 \( 1 + (-0.715 + 0.699i)T \)
29 \( 1 + (-0.450 + 0.892i)T \)
31 \( 1 + (-0.562 - 0.827i)T \)
37 \( 1 + (0.151 - 0.988i)T \)
41 \( 1 + (-0.127 + 0.991i)T \)
43 \( 1 + (-0.805 - 0.592i)T \)
47 \( 1 + (0.268 + 0.963i)T \)
53 \( 1 + (-0.432 - 0.901i)T \)
59 \( 1 + (0.771 + 0.635i)T \)
61 \( 1 + (-0.986 + 0.164i)T \)
67 \( 1 + (-0.146 + 0.989i)T \)
71 \( 1 + (-0.340 + 0.940i)T \)
73 \( 1 + (-0.528 - 0.848i)T \)
79 \( 1 + (-0.405 - 0.914i)T \)
83 \( 1 + (-0.850 + 0.526i)T \)
89 \( 1 + (0.802 + 0.596i)T \)
97 \( 1 + (-0.793 + 0.608i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.41648010882759942317650343142, −19.97987418998990857367998084733, −19.075484015074950406984029416950, −18.72309246843595552797011597686, −18.14607320621947444644018293316, −16.740167461278659506772713622277, −16.1021744755301515298250934983, −15.45329289771444387162859894727, −15.02976347229264676419182909568, −13.879772604698267070064301614396, −12.88841457875647760107174682784, −12.28111024191516003416396628681, −11.00737602210817440288974193457, −10.33092837616256539299799385775, −9.99411245997518875330044301497, −8.702469929058975260351243502278, −8.226862859054075593481452241368, −7.603754887472514028599007160584, −6.620039063859839569495136633762, −5.85218352540012463140692009872, −4.02229978394935650172439781071, −3.38324249639156252371510220006, −2.75820711490259309838382161330, −1.70119015514475916882886919456, −0.02827792532178200788135222456, 1.212862385527962563552058235228, 2.35691442288157491449833406920, 3.18455164742053990623046348856, 4.07590836521923971850838749913, 5.388746774953777615569091167329, 6.59409507187242716513656227950, 7.3571024522643310312440561865, 7.88459757810773544072302917512, 8.949957168576271162540937001923, 9.19933027682658625586997831933, 10.10647898165315969328391186608, 11.15039896173512510925941008198, 11.97001560407582444140568281442, 12.97180521378382650179894898459, 13.39263128134652604091398174257, 14.67837371861913037887784329627, 15.55301746801102685903286606696, 16.005915242473450922490207034583, 16.4524400543572906272728554424, 17.79003132564118026289407255820, 18.484641884487027850865898363565, 19.07923089111803011960907879446, 19.80019982048124903720214905821, 20.31995050243327318251779089609, 20.877556415717942451577567309309

Graph of the $Z$-function along the critical line