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Properties

Label 1-695-695.219-r0-0-0
Degree $1$
Conductor $695$
Sign $-0.710 - 0.704i$
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  + (0.576 + 0.816i)2-s + (0.0682 + 0.997i)3-s + (−0.334 + 0.942i)4-s + (−0.775 + 0.631i)6-s + (0.775 + 0.631i)7-s + (−0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (−0.917 − 0.398i)11-s + (−0.962 − 0.269i)12-s + (0.0682 + 0.997i)13-s + (−0.0682 + 0.997i)14-s + (−0.775 − 0.631i)16-s + (−0.203 + 0.979i)17-s + (−0.682 − 0.730i)18-s + (0.203 − 0.979i)19-s + ⋯
L(s)  = 1  + (0.576 + 0.816i)2-s + (0.0682 + 0.997i)3-s + (−0.334 + 0.942i)4-s + (−0.775 + 0.631i)6-s + (0.775 + 0.631i)7-s + (−0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (−0.917 − 0.398i)11-s + (−0.962 − 0.269i)12-s + (0.0682 + 0.997i)13-s + (−0.0682 + 0.997i)14-s + (−0.775 − 0.631i)16-s + (−0.203 + 0.979i)17-s + (−0.682 − 0.730i)18-s + (0.203 − 0.979i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 695 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 695 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(695\)    =    \(5 \cdot 139\)
Sign: $-0.710 - 0.704i$
Analytic conductor: \(3.22756\)
Root analytic conductor: \(3.22756\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{695} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 695,\ (0:\ ),\ -0.710 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5413362307 + 1.314901268i\)
\(L(\frac12)\) \(\approx\) \(-0.5413362307 + 1.314901268i\)
\(L(1)\) \(\approx\) \(0.6111249972 + 1.073667958i\)
\(L(1)\) \(\approx\) \(0.6111249972 + 1.073667958i\)

Euler product

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

−22.381082323544417147487945913973, −20.99455524728596864852931208556, −20.52549034361187796329561043873, −19.98439249815856939102117631072, −18.872423887930968027460832756651, −18.19535195931927636790611830608, −17.71566539392781702561320130334, −16.503609997136615444249142152248, −15.11391167763259744489663590373, −14.55481700157916787184790890551, −13.52316214360193953836036461410, −13.131542928634944248892437327556, −12.21916013365746118283813878809, −11.396535991220745249800210679194, −10.63945881661711116188518331879, −9.75757119179184104205206915619, −8.41139709307914547697122817266, −7.67988985411514361112806518344, −6.689481882801754613710424874459, −5.46637101792462101528995047762, −4.889440124331520239191914313587, −3.51196994414528592430517969544, −2.56995066238146802839073027134, −1.638443302227685501199029945431, −0.53837646122533241521434662994, 2.17886461001935630698024649658, 3.272863762063414557610287760735, 4.21774386473230965513482639818, 5.178161802209722512198138646175, 5.58186296292459681618070927952, 6.85626631128017693987504282120, 7.95909248806383342165804704080, 8.799558950155150977632040037755, 9.31391545966120266335428677520, 10.89329823514329441386023623792, 11.331686994216478368611747403465, 12.50398827997391773345933472798, 13.47499098620181898868553436496, 14.36570582120797797875070065058, 15.00392119019246901566568027218, 15.6910562528538152814754414360, 16.3683066969915286913418248786, 17.258123012679669266417806508423, 17.98207655161161471677732242098, 19.01715965682934260529210524672, 20.24647094522027901076312016230, 21.29731356929326453439723226156, 21.48065675999246665358580185632, 22.15961177904312720496722190079, 23.33080810467488553991346178923

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