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Properties

Label 1-1259-1259.1085-r1-0-0
Degree $1$
Conductor $1259$
Sign $-0.560 + 0.827i$
Analytic cond. $135.298$
Root an. cond. $135.298$
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  + (−0.978 + 0.208i)2-s + (−0.839 + 0.543i)3-s + (0.913 − 0.407i)4-s + (−0.972 + 0.232i)5-s + (0.707 − 0.706i)6-s + (0.230 − 0.973i)7-s + (−0.808 + 0.588i)8-s + (0.409 − 0.912i)9-s + (0.902 − 0.430i)10-s + (0.898 + 0.439i)11-s + (−0.545 + 0.838i)12-s + (0.485 − 0.874i)13-s + (−0.0224 + 0.999i)14-s + (0.690 − 0.723i)15-s + (0.668 − 0.744i)16-s + (−0.979 − 0.203i)17-s + ⋯
L(s)  = 1  + (−0.978 + 0.208i)2-s + (−0.839 + 0.543i)3-s + (0.913 − 0.407i)4-s + (−0.972 + 0.232i)5-s + (0.707 − 0.706i)6-s + (0.230 − 0.973i)7-s + (−0.808 + 0.588i)8-s + (0.409 − 0.912i)9-s + (0.902 − 0.430i)10-s + (0.898 + 0.439i)11-s + (−0.545 + 0.838i)12-s + (0.485 − 0.874i)13-s + (−0.0224 + 0.999i)14-s + (0.690 − 0.723i)15-s + (0.668 − 0.744i)16-s + (−0.979 − 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.560 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.560 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.560 + 0.827i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1085, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ -0.560 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09815375534 + 0.1850546156i\)
\(L(\frac12)\) \(\approx\) \(0.09815375534 + 0.1850546156i\)
\(L(1)\) \(\approx\) \(0.4382844544 + 0.03868967693i\)
\(L(1)\) \(\approx\) \(0.4382844544 + 0.03868967693i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.978 + 0.208i)T \)
3 \( 1 + (-0.839 + 0.543i)T \)
5 \( 1 + (-0.972 + 0.232i)T \)
7 \( 1 + (0.230 - 0.973i)T \)
11 \( 1 + (0.898 + 0.439i)T \)
13 \( 1 + (0.485 - 0.874i)T \)
17 \( 1 + (-0.979 - 0.203i)T \)
19 \( 1 + (0.995 + 0.0947i)T \)
23 \( 1 + (-0.537 + 0.843i)T \)
29 \( 1 + (-0.985 - 0.169i)T \)
31 \( 1 + (-0.574 - 0.818i)T \)
37 \( 1 + (-0.999 - 0.00499i)T \)
41 \( 1 + (-0.372 - 0.927i)T \)
43 \( 1 + (-0.395 + 0.918i)T \)
47 \( 1 + (-0.288 - 0.957i)T \)
53 \( 1 + (0.990 + 0.139i)T \)
59 \( 1 + (-0.307 - 0.951i)T \)
61 \( 1 + (-0.553 + 0.832i)T \)
67 \( 1 + (0.0723 + 0.997i)T \)
71 \( 1 + (-0.117 - 0.993i)T \)
73 \( 1 + (-0.00749 + 0.999i)T \)
79 \( 1 + (-0.822 - 0.568i)T \)
83 \( 1 + (0.739 - 0.673i)T \)
89 \( 1 + (0.884 + 0.465i)T \)
97 \( 1 + (-0.656 - 0.753i)T \)
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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

−20.28355287543358396300300149891, −19.672710958795559889307976075079, −18.83341829164351185744810879485, −18.4794133549795803635798273868, −17.71115880851672448656989255378, −16.76152273788952494040514306530, −16.22224151382447036512265596803, −15.64213264710385012239596648718, −14.6487577004032234116190050841, −13.41814511019000642237051749738, −12.23430500160432145574246741292, −12.01308471780339904736759884047, −11.25100947409029351705240437423, −10.790365440830205956634759072570, −9.35204275169974588980635103762, −8.73527590936595973151785671159, −8.061816606695357916630945219066, −7.01216169746233752981493715404, −6.50103052494418408287763485176, −5.5072409844036545081981471275, −4.35019012890826879332065796295, −3.30032898779220292443278552294, −1.97737724550356967735473547783, −1.282656950797198095668415937502, −0.10715060801011074868074576413, 0.65944753829137261330127012635, 1.64836075005313388213632005040, 3.40677196780729490267931580488, 3.973147593131181451445029684753, 5.10580410768678008136031840544, 6.08411906643444294942154589599, 7.104958890974323695563860941037, 7.41671354880618743803516510370, 8.52226359754410613971216992422, 9.47415063710273704892921364447, 10.21346670302247323900196186152, 10.98466721370419898615713082684, 11.49493385253939666403936285254, 12.105722200856802234017662857222, 13.39392389358247333806676995820, 14.64267448104221319734364492960, 15.274802989913602428343990385435, 15.923612658412924030313054625988, 16.580053424541829922275683384387, 17.37172933830036387478344278512, 17.88739228896627690076045793566, 18.647546050960079046257879681870, 19.80064800532687325167705614241, 20.18718074640479913943329573472, 20.75736754643802233728183888633

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