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Properties

Label 2-231-33.32-c1-0-10
Degree $2$
Conductor $231$
Sign $0.685 + 0.728i$
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  − 2.30·2-s + (−0.220 + 1.71i)3-s + 3.29·4-s − 1.52i·5-s + (0.507 − 3.95i)6-s − i·7-s − 2.98·8-s + (−2.90 − 0.757i)9-s + 3.50i·10-s + (−2.56 − 2.10i)11-s + (−0.726 + 5.66i)12-s − 2.11i·13-s + 2.30i·14-s + (2.62 + 0.336i)15-s + 0.270·16-s + 7.10·17-s + ⋯
L(s)  = 1  − 1.62·2-s + (−0.127 + 0.991i)3-s + 1.64·4-s − 0.682i·5-s + (0.207 − 1.61i)6-s − 0.377i·7-s − 1.05·8-s + (−0.967 − 0.252i)9-s + 1.10i·10-s + (−0.772 − 0.635i)11-s + (−0.209 + 1.63i)12-s − 0.586i·13-s + 0.615i·14-s + (0.676 + 0.0867i)15-s + 0.0677·16-s + 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.685 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418824 - 0.181052i\)
\(L(\frac12)\) \(\approx\) \(0.418824 - 0.181052i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.220 - 1.71i)T \)
7 \( 1 + iT \)
11 \( 1 + (2.56 + 2.10i)T \)
good2 \( 1 + 2.30T + 2T^{2} \)
5 \( 1 + 1.52iT - 5T^{2} \)
13 \( 1 + 2.11iT - 13T^{2} \)
17 \( 1 - 7.10T + 17T^{2} \)
19 \( 1 + 1.02iT - 19T^{2} \)
23 \( 1 + 7.02iT - 23T^{2} \)
29 \( 1 - 0.896T + 29T^{2} \)
31 \( 1 + 2.10T + 31T^{2} \)
37 \( 1 - 6.44T + 37T^{2} \)
41 \( 1 + 4.95T + 41T^{2} \)
43 \( 1 + 4.22iT - 43T^{2} \)
47 \( 1 + 1.52iT - 47T^{2} \)
53 \( 1 - 12.9iT - 53T^{2} \)
59 \( 1 + 4.69iT - 59T^{2} \)
61 \( 1 + 1.83iT - 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 - 7.36iT - 71T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 + 9.00iT - 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 14.6iT - 89T^{2} \)
97 \( 1 + 2.68T + 97T^{2} \)
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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

−11.62816758107450406052672831238, −10.45628974211660867008757080483, −10.31135754149082457233136058368, −9.119832065369142790041626988059, −8.409788049817269528227196132090, −7.59586971264167357116911622882, −5.96392596152595595104727289796, −4.76573458396477325443095806949, −3.01741916518261471598394534478, −0.67556131346063491001811951778, 1.55186739371723893133931517696, 2.86864163341020050304143253075, 5.52708560573349056551571786507, 6.78116645425318942014598867160, 7.54693815214865434660498024768, 8.188156822289942633156755394874, 9.434171310713313440747474918363, 10.22311241412225200923223631444, 11.26312503636567398111773599708, 11.96591874983502679371032075030

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