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Properties

Label 2-231-7.2-c1-0-8
Degree $2$
Conductor $231$
Sign $0.947 + 0.318i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.207 + 0.358i)2-s + (−0.5 − 0.866i)3-s + (0.914 + 1.58i)4-s + (1 − 1.73i)5-s + 0.414·6-s + (1 − 2.44i)7-s − 1.58·8-s + (−0.499 + 0.866i)9-s + (0.414 + 0.717i)10-s + (−0.5 − 0.866i)11-s + (0.914 − 1.58i)12-s + 4.82·13-s + (0.671 + 0.866i)14-s − 1.99·15-s + (−1.49 + 2.59i)16-s + (0.792 + 1.37i)17-s + ⋯
L(s)  = 1  + (−0.146 + 0.253i)2-s + (−0.288 − 0.499i)3-s + (0.457 + 0.791i)4-s + (0.447 − 0.774i)5-s + 0.169·6-s + (0.377 − 0.925i)7-s − 0.560·8-s + (−0.166 + 0.288i)9-s + (0.130 + 0.226i)10-s + (−0.150 − 0.261i)11-s + (0.263 − 0.457i)12-s + 1.33·13-s + (0.179 + 0.231i)14-s − 0.516·15-s + (−0.374 + 0.649i)16-s + (0.192 + 0.333i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25193 - 0.204748i\)
\(L(\frac12)\) \(\approx\) \(1.25193 - 0.204748i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1 + 2.44i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.207 - 0.358i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 4.82T + 13T^{2} \)
17 \( 1 + (-0.792 - 1.37i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.621 + 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.24T + 29T^{2} \)
31 \( 1 + (-2.82 - 4.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.74 - 6.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + (2.08 - 3.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.41 - 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.32 + 2.30i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.41 - 7.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.82T + 71T^{2} \)
73 \( 1 + (5.82 + 10.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.65 + 8.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + (7.07 - 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.48T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.20162576041825840440226005425, −11.24634514930819375254882363801, −10.43790907474413634001507395898, −8.788763174015576871125959670029, −8.262619628631563824459119170586, −7.09896891494236227805755853127, −6.24618221472670571684636228146, −4.87719032712782475385318253583, −3.38002153120382015028702296787, −1.40217914940343582907411199075, 1.88067351098416034493447782242, 3.31500909229642073910930122497, 5.26401539996657143034697523382, 5.92644459684163194675063749848, 6.97475293887977812143633511054, 8.594567773615516353443593394175, 9.616149023428074746990932136839, 10.34732051566043573864146424169, 11.32262551965113958101661023738, 11.69517710496901999554057873019

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