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Properties

Label 1-1259-1259.1017-r0-0-0
Degree $1$
Conductor $1259$
Sign $0.251 + 0.967i$
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  + (−0.485 + 0.874i)2-s + (−0.902 + 0.430i)3-s + (−0.528 − 0.848i)4-s + (0.675 + 0.737i)5-s + (0.0623 − 0.998i)6-s + (−0.578 + 0.815i)7-s + (0.998 − 0.0499i)8-s + (0.630 − 0.776i)9-s + (−0.972 + 0.232i)10-s + (−0.742 − 0.669i)11-s + (0.842 + 0.539i)12-s + (−0.977 − 0.213i)13-s + (−0.432 − 0.901i)14-s + (−0.926 − 0.375i)15-s + (−0.441 + 0.897i)16-s + (−0.686 − 0.727i)17-s + ⋯
L(s)  = 1  + (−0.485 + 0.874i)2-s + (−0.902 + 0.430i)3-s + (−0.528 − 0.848i)4-s + (0.675 + 0.737i)5-s + (0.0623 − 0.998i)6-s + (−0.578 + 0.815i)7-s + (0.998 − 0.0499i)8-s + (0.630 − 0.776i)9-s + (−0.972 + 0.232i)10-s + (−0.742 − 0.669i)11-s + (0.842 + 0.539i)12-s + (−0.977 − 0.213i)13-s + (−0.432 − 0.901i)14-s + (−0.926 − 0.375i)15-s + (−0.441 + 0.897i)16-s + (−0.686 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.251 + 0.967i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ 0.251 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4688385901 + 0.3625151883i\)
\(L(\frac12)\) \(\approx\) \(0.4688385901 + 0.3625151883i\)
\(L(1)\) \(\approx\) \(0.4797618152 + 0.3015188912i\)
\(L(1)\) \(\approx\) \(0.4797618152 + 0.3015188912i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.485 + 0.874i)T \)
3 \( 1 + (-0.902 + 0.430i)T \)
5 \( 1 + (0.675 + 0.737i)T \)
7 \( 1 + (-0.578 + 0.815i)T \)
11 \( 1 + (-0.742 - 0.669i)T \)
13 \( 1 + (-0.977 - 0.213i)T \)
17 \( 1 + (-0.686 - 0.727i)T \)
19 \( 1 + (0.0324 - 0.999i)T \)
23 \( 1 + (0.997 + 0.0698i)T \)
29 \( 1 + (-0.594 + 0.803i)T \)
31 \( 1 + (-0.679 - 0.733i)T \)
37 \( 1 + (0.968 + 0.247i)T \)
41 \( 1 + (-0.967 + 0.251i)T \)
43 \( 1 + (-0.0673 - 0.997i)T \)
47 \( 1 + (0.454 + 0.890i)T \)
53 \( 1 + (0.758 + 0.651i)T \)
59 \( 1 + (0.995 + 0.0997i)T \)
61 \( 1 + (0.481 + 0.876i)T \)
67 \( 1 + (0.887 - 0.461i)T \)
71 \( 1 + (-0.915 - 0.402i)T \)
73 \( 1 + (-0.930 - 0.365i)T \)
79 \( 1 + (0.363 - 0.931i)T \)
83 \( 1 + (0.739 + 0.673i)T \)
89 \( 1 + (0.614 - 0.788i)T \)
97 \( 1 + (0.288 - 0.957i)T \)
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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

−20.825705082805712032280354680934, −20.11345078545721861821842902985, −19.396146812544278556057342029839, −18.591244541615122522170440994, −17.76463968191331571208144193788, −17.16178764619683016743906303995, −16.74410801668969615485816841005, −16.00157588456403790577916043933, −14.55497460434936611844052161686, −13.28366728722637159581704870127, −13.053830208837680326030229535408, −12.45081055592605072132609125294, −11.57871141984041043500966050177, −10.55064378187373733758016549143, −10.09504541409856566177718201962, −9.439725549929160368090235213996, −8.27204254977821301733137732808, −7.40929817667641594177997541488, −6.67396328748044023710273646266, −5.447580634652439098384616434800, −4.71203607928656397522836886416, −3.88152488660893956876871448081, −2.380340334811214041472034635805, −1.71767248511638898047034192979, −0.636084395801863387165053010977, 0.53348354302695809396945246096, 2.225150422875968101741459432348, 3.16861099726054957501965108136, 4.73510861917809741046974008866, 5.40666328638268530975842073753, 5.94771125677305082095448747811, 6.88553070740139466803907498964, 7.34747493516848181698730178279, 8.918389671054329614887452763925, 9.3514589010964787154707506088, 10.17480584199920241943276935941, 10.87722465003624614912261523818, 11.60448809997975852750569906211, 12.989005170904150530190563154155, 13.414864658811029451730755452248, 14.74885354577602665930683865725, 15.18550615345291258687437126466, 15.89770873724229095001120261691, 16.6772510234936135954996284349, 17.34801469535053987971688167816, 18.16541624296324844592579140212, 18.53480415781784103775235247596, 19.31512797626255196315190926037, 20.48589823380532872282997277916, 21.69467266449807559487300399544

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