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Properties

Label 2-231-77.76-c1-0-12
Degree $2$
Conductor $231$
Sign $0.0403 + 0.999i$
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  − 1.13i·2-s + i·3-s + 0.714·4-s − 2.77i·5-s + 1.13·6-s + (−2.60 − 0.474i)7-s − 3.07i·8-s − 9-s − 3.14·10-s + (3.23 − 0.726i)11-s + 0.714i·12-s + 1.20·13-s + (−0.538 + 2.95i)14-s + 2.77·15-s − 2.05·16-s + 6.01·17-s + ⋯
L(s)  = 1  − 0.801i·2-s + 0.577i·3-s + 0.357·4-s − 1.24i·5-s + 0.462·6-s + (−0.983 − 0.179i)7-s − 1.08i·8-s − 0.333·9-s − 0.994·10-s + (0.975 − 0.219i)11-s + 0.206i·12-s + 0.333·13-s + (−0.143 + 0.788i)14-s + 0.716·15-s − 0.514·16-s + 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0403 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0403 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.0403 + 0.999i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.0403 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953771 - 0.916017i\)
\(L(\frac12)\) \(\approx\) \(0.953771 - 0.916017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (2.60 + 0.474i)T \)
11 \( 1 + (-3.23 + 0.726i)T \)
good2 \( 1 + 1.13iT - 2T^{2} \)
5 \( 1 + 2.77iT - 5T^{2} \)
13 \( 1 - 1.20T + 13T^{2} \)
17 \( 1 - 6.01T + 17T^{2} \)
19 \( 1 + 2.01T + 19T^{2} \)
23 \( 1 + 5.90T + 23T^{2} \)
29 \( 1 - 6.40iT - 29T^{2} \)
31 \( 1 - 6.11iT - 31T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 - 0.503T + 41T^{2} \)
43 \( 1 - 9.37iT - 43T^{2} \)
47 \( 1 + 7.12iT - 47T^{2} \)
53 \( 1 + 1.42T + 53T^{2} \)
59 \( 1 - 5.81iT - 59T^{2} \)
61 \( 1 - 4.53T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 - 8.02T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 - 16.3iT - 79T^{2} \)
83 \( 1 + 0.143T + 83T^{2} \)
89 \( 1 - 3.07iT - 89T^{2} \)
97 \( 1 + 3.26iT - 97T^{2} \)
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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

−12.24225423798305846984241278511, −10.98508881715089197865904147724, −9.993523049108857362562966880776, −9.377709711486185334874972817802, −8.338403623011277472142369342429, −6.76707775318542327599585953285, −5.66518720600963143049716806083, −4.13990112292016204762271669199, −3.25223240147160240677545032411, −1.23414832611518512301911100706, 2.33042899740542984839461411731, 3.62185558226779281784880524804, 5.96227900942277435010316807962, 6.31832038283631782407204528099, 7.24909718579789557574688320157, 8.062459096183397132115568939680, 9.517658045498525750836424807657, 10.51814882907408294832517091473, 11.62673648528393338880967525320, 12.27609256048286730081365444015

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