Properties

Label 2-231-11.3-c3-0-21
Degree $2$
Conductor $231$
Sign $0.711 + 0.702i$
Analytic cond. $13.6294$
Root an. cond. $3.69180$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.992 + 0.721i)2-s + (−0.927 − 2.85i)3-s + (−2.00 + 6.17i)4-s + (2.71 + 1.97i)5-s + (2.97 + 2.16i)6-s + (2.16 − 6.65i)7-s + (−5.49 − 16.9i)8-s + (−7.28 + 5.29i)9-s − 4.12·10-s + (−36.1 − 4.60i)11-s + 19.4·12-s + (35.9 − 26.1i)13-s + (2.65 + 8.16i)14-s + (3.11 − 9.58i)15-s + (−24.3 − 17.7i)16-s + (26.8 + 19.5i)17-s + ⋯
L(s)  = 1  + (−0.350 + 0.254i)2-s + (−0.178 − 0.549i)3-s + (−0.250 + 0.772i)4-s + (0.243 + 0.176i)5-s + (0.202 + 0.147i)6-s + (0.116 − 0.359i)7-s + (−0.242 − 0.747i)8-s + (−0.269 + 0.195i)9-s − 0.130·10-s + (−0.991 − 0.126i)11-s + 0.468·12-s + (0.766 − 0.556i)13-s + (0.0506 + 0.155i)14-s + (0.0536 − 0.165i)15-s + (−0.380 − 0.276i)16-s + (0.383 + 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(13.6294\)
Root analytic conductor: \(3.69180\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :3/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.03234 - 0.423584i\)
\(L(\frac12)\) \(\approx\) \(1.03234 - 0.423584i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.927 + 2.85i)T \)
7 \( 1 + (-2.16 + 6.65i)T \)
11 \( 1 + (36.1 + 4.60i)T \)
good2 \( 1 + (0.992 - 0.721i)T + (2.47 - 7.60i)T^{2} \)
5 \( 1 + (-2.71 - 1.97i)T + (38.6 + 118. i)T^{2} \)
13 \( 1 + (-35.9 + 26.1i)T + (678. - 2.08e3i)T^{2} \)
17 \( 1 + (-26.8 - 19.5i)T + (1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (-21.1 - 65.1i)T + (-5.54e3 + 4.03e3i)T^{2} \)
23 \( 1 - 65.7T + 1.21e4T^{2} \)
29 \( 1 + (-71.7 + 220. i)T + (-1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (-222. + 161. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-28.4 + 87.6i)T + (-4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (72.0 + 221. i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 - 312.T + 7.95e4T^{2} \)
47 \( 1 + (107. + 329. i)T + (-8.39e4 + 6.10e4i)T^{2} \)
53 \( 1 + (589. - 428. i)T + (4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-215. + 664. i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (-509. - 370. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 - 1.09e3T + 3.00e5T^{2} \)
71 \( 1 + (125. + 91.0i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (132. - 408. i)T + (-3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (216. - 157. i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (412. + 299. i)T + (1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 - 336.T + 7.04e5T^{2} \)
97 \( 1 + (844. - 613. i)T + (2.82e5 - 8.68e5i)T^{2} \)
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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

−11.73833186568670048787825574289, −10.59666332297843147032453725847, −9.698763840605597378256135957508, −8.084720218668722568735417639421, −8.049155543698125403573515992906, −6.68574923210715411631755735674, −5.62985483899360334683928794269, −4.02036405449223873756691852379, −2.64800920015042330910806541821, −0.61478326401253917265736106591, 1.21893030690451184679945851773, 2.89551948313642977045980857811, 4.78484584843483998975453201971, 5.40519749643122449454399623835, 6.64248839427536515585785563829, 8.289786305467978316028018739532, 9.157000944692769059148598593778, 9.900863209099034932011388646940, 10.86771511601175594970209825907, 11.47032874445879945562619127569

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