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Properties

Label 1-31e2-961.5-r0-0-0
Degree $1$
Conductor $961$
Sign $-0.945 + 0.324i$
Analytic cond. $4.46286$
Root an. cond. $4.46286$
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  + (0.151 + 0.988i)2-s + (−0.985 + 0.168i)3-s + (−0.954 + 0.299i)4-s + (0.470 + 0.882i)5-s + (−0.315 − 0.948i)6-s + (−0.378 + 0.925i)7-s + (−0.440 − 0.897i)8-s + (0.943 − 0.331i)9-s + (−0.801 + 0.598i)10-s + (0.409 − 0.912i)11-s + (0.890 − 0.455i)12-s + (−0.184 − 0.982i)13-s + (−0.972 − 0.234i)14-s + (−0.612 − 0.790i)15-s + (0.820 − 0.571i)16-s + (0.963 − 0.266i)17-s + ⋯
L(s)  = 1  + (0.151 + 0.988i)2-s + (−0.985 + 0.168i)3-s + (−0.954 + 0.299i)4-s + (0.470 + 0.882i)5-s + (−0.315 − 0.948i)6-s + (−0.378 + 0.925i)7-s + (−0.440 − 0.897i)8-s + (0.943 − 0.331i)9-s + (−0.801 + 0.598i)10-s + (0.409 − 0.912i)11-s + (0.890 − 0.455i)12-s + (−0.184 − 0.982i)13-s + (−0.972 − 0.234i)14-s + (−0.612 − 0.790i)15-s + (0.820 − 0.571i)16-s + (0.963 − 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.945 + 0.324i$
Analytic conductor: \(4.46286\)
Root analytic conductor: \(4.46286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 961,\ (0:\ ),\ -0.945 + 0.324i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1746488658 + 1.048279641i\)
\(L(\frac12)\) \(\approx\) \(0.1746488658 + 1.048279641i\)
\(L(1)\) \(\approx\) \(0.5934442960 + 0.6262152802i\)
\(L(1)\) \(\approx\) \(0.5934442960 + 0.6262152802i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (0.151 + 0.988i)T \)
3 \( 1 + (-0.985 + 0.168i)T \)
5 \( 1 + (0.470 + 0.882i)T \)
7 \( 1 + (-0.378 + 0.925i)T \)
11 \( 1 + (0.409 - 0.912i)T \)
13 \( 1 + (-0.184 - 0.982i)T \)
17 \( 1 + (0.963 - 0.266i)T \)
19 \( 1 + (0.0168 + 0.999i)T \)
23 \( 1 + (0.528 + 0.848i)T \)
29 \( 1 + (0.528 + 0.848i)T \)
37 \( 1 + (0.943 + 0.331i)T \)
41 \( 1 + (0.997 + 0.0675i)T \)
43 \( 1 + (-0.801 - 0.598i)T \)
47 \( 1 + (0.528 + 0.848i)T \)
53 \( 1 + (0.638 + 0.769i)T \)
59 \( 1 + (0.943 + 0.331i)T \)
61 \( 1 + (-0.0506 - 0.998i)T \)
67 \( 1 + (-0.713 - 0.701i)T \)
71 \( 1 + (0.0168 - 0.999i)T \)
73 \( 1 + (0.283 + 0.959i)T \)
79 \( 1 + (0.283 - 0.959i)T \)
83 \( 1 + (-0.931 - 0.363i)T \)
89 \( 1 + (-0.612 + 0.790i)T \)
97 \( 1 + (-0.954 - 0.299i)T \)
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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

−21.30370380662146131597478816229, −20.814666391124886810557767071563, −19.78697594138964627264294871043, −19.300089147689257133739873010709, −18.19049108212052970343281780033, −17.467055362486352963069608598695, −16.87296033154158624960459521661, −16.3025919607922290584865074730, −14.8740487628452758142565362559, −13.883616723337449556545634017317, −13.1296085438305311105191718062, −12.560298154795233716634717693970, −11.83584606201167305635525647681, −11.0605567982531228127238866118, −9.885041128483136200807527147871, −9.81593841822879478990999634495, −8.61984299867672273832875533269, −7.30582783587374569240584293191, −6.4181667777133849944968957216, −5.3940643578141649642386818404, −4.43410282427067239631836302637, −4.1670671351329442075074418648, −2.45344898083576643273430923330, −1.40191910112241227747403010989, −0.65175923093249747189748425555, 1.10086397717939668140988995794, 2.97388347151868243276063772648, 3.65100499962273482790499813902, 5.13582970058274032465675554454, 5.78678328103876678771266726460, 6.14312716786456541414520335477, 7.13679916011122866201611551193, 8.01324006402563168742753233465, 9.22564800192390316521823093770, 9.89683276362114849910779244766, 10.74020310714393303630838691577, 11.83074320825428385238425094300, 12.54263077623938376970830841021, 13.42504084252710167837678136850, 14.40932546956902421542239876790, 15.06960709942949681356141417547, 15.845352653373814697578503363537, 16.57179804566471445080754420223, 17.2815356230211661987674319680, 18.16221661390100328814316711533, 18.56690354763352432875042337394, 19.31565680954197258497563426192, 21.08670379940600271041940935164, 21.72274554970500952517675272238, 22.21089326971145063016295587145

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