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Properties

Label 1-695-695.242-r0-0-0
Degree $1$
Conductor $695$
Sign $0.585 + 0.810i$
Analytic cond. $3.22756$
Root an. cond. $3.22756$
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  + (−0.631 + 0.775i)2-s + (0.269 + 0.962i)3-s + (−0.203 − 0.979i)4-s + (−0.917 − 0.398i)6-s + (−0.398 − 0.917i)7-s + (0.887 + 0.460i)8-s + (−0.854 + 0.519i)9-s + (−0.0682 + 0.997i)11-s + (0.887 − 0.460i)12-s + (−0.269 − 0.962i)13-s + (0.962 + 0.269i)14-s + (−0.917 + 0.398i)16-s + (0.730 − 0.682i)17-s + (0.136 − 0.990i)18-s + (0.682 + 0.730i)19-s + ⋯
L(s)  = 1  + (−0.631 + 0.775i)2-s + (0.269 + 0.962i)3-s + (−0.203 − 0.979i)4-s + (−0.917 − 0.398i)6-s + (−0.398 − 0.917i)7-s + (0.887 + 0.460i)8-s + (−0.854 + 0.519i)9-s + (−0.0682 + 0.997i)11-s + (0.887 − 0.460i)12-s + (−0.269 − 0.962i)13-s + (0.962 + 0.269i)14-s + (−0.917 + 0.398i)16-s + (0.730 − 0.682i)17-s + (0.136 − 0.990i)18-s + (0.682 + 0.730i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 695 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 695 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(695\)    =    \(5 \cdot 139\)
Sign: $0.585 + 0.810i$
Analytic conductor: \(3.22756\)
Root analytic conductor: \(3.22756\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{695} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 695,\ (0:\ ),\ 0.585 + 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8590522630 + 0.4393322339i\)
\(L(\frac12)\) \(\approx\) \(0.8590522630 + 0.4393322339i\)
\(L(1)\) \(\approx\) \(0.7257000593 + 0.3551658276i\)
\(L(1)\) \(\approx\) \(0.7257000593 + 0.3551658276i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.631 + 0.775i)T \)
3 \( 1 + (0.269 + 0.962i)T \)
7 \( 1 + (-0.398 - 0.917i)T \)
11 \( 1 + (-0.0682 + 0.997i)T \)
13 \( 1 + (-0.269 - 0.962i)T \)
17 \( 1 + (0.730 - 0.682i)T \)
19 \( 1 + (0.682 + 0.730i)T \)
23 \( 1 + (-0.398 - 0.917i)T \)
29 \( 1 + (-0.203 - 0.979i)T \)
31 \( 1 + (0.854 + 0.519i)T \)
37 \( 1 + (0.942 - 0.334i)T \)
41 \( 1 + (-0.990 + 0.136i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.816 - 0.576i)T \)
53 \( 1 + (0.631 - 0.775i)T \)
59 \( 1 + (0.203 - 0.979i)T \)
61 \( 1 + (0.990 + 0.136i)T \)
67 \( 1 + (0.997 - 0.0682i)T \)
71 \( 1 + (-0.576 + 0.816i)T \)
73 \( 1 + (0.887 + 0.460i)T \)
79 \( 1 + (0.990 + 0.136i)T \)
83 \( 1 + (0.997 + 0.0682i)T \)
89 \( 1 + (0.334 - 0.942i)T \)
97 \( 1 - iT \)
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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

−22.12142699765943898691871844809, −21.79252031285534012768115819066, −20.79777575638856731069522030713, −19.80379362182848125135511365045, −19.127688623423561044077836482933, −18.737645409903466512834378906, −17.96073448477685738385886479921, −17.00947054878448255991257609342, −16.26195077376943781944077339838, −15.11072638778803508301924665269, −13.850500114956073344255475554525, −13.375873825260509140589981612769, −12.24963927655709815307970088007, −11.867244518477215607820884738382, −11.010456326746621300144393093165, −9.68927259811377184868396633014, −8.99494112317820993313643046382, −8.28960215034285488074502963539, −7.38389267708712038296300480977, −6.396602621953157684513494328179, −5.37733354629706992786179853221, −3.70826148043054147550880505388, −2.90001643524460208832363699841, −2.03344818437619287622558063423, −0.9505865233860190215937139486, 0.7407214541687172854863543218, 2.43291406596495033585744537251, 3.71186649077687938363116256419, 4.70550090982480885574764328222, 5.47450423747950092286788678622, 6.623581030715645221655356868942, 7.68230745636983373929255454348, 8.18427688535878087638547386079, 9.57825859196840647003958475046, 9.966671738647204557423206356846, 10.49917040580901960677739317176, 11.74515787821189079506307332890, 13.08250774707333031934437281659, 14.10353549399223773391933468527, 14.661472449056732257331187132349, 15.55651798032349364408618284364, 16.22921405986743393176554863054, 16.925300195309084291573410702272, 17.65187205767048418536079952505, 18.60120075074300176444746298096, 19.70322058556297874078689932737, 20.281083909558038556785854169629, 20.81406765567421490732615518049, 22.33589807988907214833643404964, 22.866616280407020274022227357556

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