Properties

Label 1-967-967.44-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.714 + 0.699i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.807 + 0.589i)2-s + (−0.724 + 0.689i)3-s + (0.304 − 0.952i)4-s + (−0.347 − 0.937i)5-s + (0.177 − 0.984i)6-s + (−0.982 + 0.187i)7-s + (0.316 + 0.948i)8-s + (0.0487 − 0.998i)9-s + (0.833 + 0.552i)10-s + (0.811 − 0.584i)11-s + (0.436 + 0.899i)12-s + (−0.911 + 0.410i)13-s + (0.682 − 0.730i)14-s + (0.898 + 0.439i)15-s + (−0.815 − 0.579i)16-s + (−0.822 + 0.568i)17-s + ⋯
L(s)  = 1  + (−0.807 + 0.589i)2-s + (−0.724 + 0.689i)3-s + (0.304 − 0.952i)4-s + (−0.347 − 0.937i)5-s + (0.177 − 0.984i)6-s + (−0.982 + 0.187i)7-s + (0.316 + 0.948i)8-s + (0.0487 − 0.998i)9-s + (0.833 + 0.552i)10-s + (0.811 − 0.584i)11-s + (0.436 + 0.899i)12-s + (−0.911 + 0.410i)13-s + (0.682 − 0.730i)14-s + (0.898 + 0.439i)15-s + (−0.815 − 0.579i)16-s + (−0.822 + 0.568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.714 + 0.699i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.714 + 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06161922786 + 0.1509577690i\)
\(L(\frac12)\) \(\approx\) \(0.06161922786 + 0.1509577690i\)
\(L(1)\) \(\approx\) \(0.3923261582 + 0.08341672142i\)
\(L(1)\) \(\approx\) \(0.3923261582 + 0.08341672142i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.807 + 0.589i)T \)
3 \( 1 + (-0.724 + 0.689i)T \)
5 \( 1 + (-0.347 - 0.937i)T \)
7 \( 1 + (-0.982 + 0.187i)T \)
11 \( 1 + (0.811 - 0.584i)T \)
13 \( 1 + (-0.911 + 0.410i)T \)
17 \( 1 + (-0.822 + 0.568i)T \)
19 \( 1 + (-0.741 - 0.670i)T \)
23 \( 1 + (0.00975 - 0.999i)T \)
29 \( 1 + (0.389 - 0.921i)T \)
31 \( 1 + (-0.767 + 0.641i)T \)
37 \( 1 + (0.643 - 0.765i)T \)
41 \( 1 + (0.460 - 0.887i)T \)
43 \( 1 + (-0.953 - 0.300i)T \)
47 \( 1 + (-0.618 - 0.785i)T \)
53 \( 1 + (-0.158 + 0.987i)T \)
59 \( 1 + (0.571 + 0.820i)T \)
61 \( 1 + (0.538 + 0.842i)T \)
67 \( 1 + (0.938 - 0.344i)T \)
71 \( 1 + (-0.107 + 0.994i)T \)
73 \( 1 + (-0.158 - 0.987i)T \)
79 \( 1 + (-0.927 + 0.374i)T \)
83 \( 1 + (-0.658 - 0.752i)T \)
89 \( 1 + (-0.171 + 0.985i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−21.82006789996166339045207022061, −20.20126598639639457010841600958, −19.64842418271241534683261437650, −19.157954322147973188872881824967, −18.31321172083688095000959957290, −17.69722884496385793300386779434, −16.9585768874195944503830002566, −16.23643740621994883295608651848, −15.28950126422826580883510991668, −14.218522578414916854165513376485, −12.98024942120576058932566791749, −12.60847873792972168753024008781, −11.54291417669167169786001525710, −11.1935946116413358516790848250, −10.005577644996224642241820679395, −9.70716889960180580618709344055, −8.28593305079359121614344405712, −7.326078489719015463050612015957, −6.871320810116504849947511894037, −6.17410211040702803732944254118, −4.59089661780416708072263691186, −3.49097598805857766910691738122, −2.57836671474816831633766209230, −1.63082571339910874796090799130, −0.1466848157908717079389243498, 0.773979778355047853341757380442, 2.300193868871150008501537263428, 3.92182099459907997897832857208, 4.64211905084303563932466026257, 5.65571013564025690923823940785, 6.422515679607513981642093349364, 7.0871255396324825942968000592, 8.627746029785839057035111054151, 8.95280844487292370182251783762, 9.74240173601306417759227005764, 10.595246329569373330891109623258, 11.49874730679859449207981814132, 12.25281047652976895891075645271, 13.17007406488153041189091318863, 14.48223409255229385532497940776, 15.28134531686791636753187025772, 15.95464998855233481807286093722, 16.67894808729317760179497782187, 16.98116303873867732041103753336, 17.8088944636797338902456270111, 19.00403118166773039546433207930, 19.63779558250609125645134117140, 20.1558166241378352250270188970, 21.39099911288778082438914565115, 22.03822417060271130632759697990

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