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Properties

Label 1-837-837.364-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.949 - 0.314i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (0.961 + 0.275i)2-s + (0.848 + 0.529i)4-s + (0.173 + 0.984i)5-s + (−0.374 + 0.927i)7-s + (0.669 + 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.990 + 0.139i)11-s + (−0.961 + 0.275i)13-s + (−0.615 + 0.788i)14-s + (0.438 + 0.898i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (−0.374 + 0.927i)20-s + (−0.990 − 0.139i)22-s + (0.374 + 0.927i)23-s + ⋯
L(s)  = 1  + (0.961 + 0.275i)2-s + (0.848 + 0.529i)4-s + (0.173 + 0.984i)5-s + (−0.374 + 0.927i)7-s + (0.669 + 0.743i)8-s + (−0.104 + 0.994i)10-s + (−0.990 + 0.139i)11-s + (−0.961 + 0.275i)13-s + (−0.615 + 0.788i)14-s + (0.438 + 0.898i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (−0.374 + 0.927i)20-s + (−0.990 − 0.139i)22-s + (0.374 + 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.949 - 0.314i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.949 - 0.314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4059930114 + 2.517505708i\)
\(L(\frac12)\) \(\approx\) \(-0.4059930114 + 2.517505708i\)
\(L(1)\) \(\approx\) \(1.310675749 + 1.012077942i\)
\(L(1)\) \(\approx\) \(1.310675749 + 1.012077942i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.961 + 0.275i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.374 + 0.927i)T \)
11 \( 1 + (-0.990 + 0.139i)T \)
13 \( 1 + (-0.961 + 0.275i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.374 + 0.927i)T \)
29 \( 1 + (-0.961 - 0.275i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.559 + 0.829i)T \)
43 \( 1 + (0.719 - 0.694i)T \)
47 \( 1 + (-0.997 - 0.0697i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.719 - 0.694i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (-0.0348 - 0.999i)T \)
83 \( 1 + (-0.961 - 0.275i)T \)
89 \( 1 + (0.978 + 0.207i)T \)
97 \( 1 + (0.990 - 0.139i)T \)
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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

−21.19406037654899267944132407972, −20.93054703752302822335786309370, −20.060447464657627768862786351260, −19.513053089522744940515694472341, −18.45663979268030288476539885210, −17.23986981431436980755222911922, −16.390549039010503527831098501240, −16.07442083608741432859299663438, −14.806206104662072713612640478, −14.13145308114538509565127648081, −13.16616096325342695874482901515, −12.74203845983076457048408650117, −12.00541506506513014632318264178, −10.823564467719154358384622608435, −10.12376284596711230526312373406, −9.38038363679923694729297884167, −7.819035206904160814735591540847, −7.37687747210980513129222171583, −6.00415494508805694235473855469, −5.28157251811705077649062579361, −4.53570015810238049253765611933, −3.54241686371461858672145709354, −2.580497432102644011019326767016, −1.3187295521926239165758604794, −0.36205137601650407881449059229, 1.89309321005800673401410973604, 2.8779428654633530344339416752, 3.24407757183789176716543764250, 4.78483935199155319878814847989, 5.53734372318622693031898618133, 6.249433103888831398224607086054, 7.41606669698000019242998821720, 7.714750296343942024724122330629, 9.34774886119193300599998879986, 10.09005580794627326487219240185, 11.2223497774403474769674896960, 11.84480751509049989368169871136, 12.72195946696418349640871908501, 13.53517506205356656294745648139, 14.372549340961988873097518325721, 15.13729365791155095009937623505, 15.613111803351579016023537550632, 16.55830890716694095537055135439, 17.548519512737173010274724163147, 18.474430434586955716579051798241, 19.1343736727420393810911505781, 20.125488097479148874017262526795, 21.189830393902792594216277089748, 21.69153757246222229612753561559, 22.38675790373284149778631660159

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