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Properties

Label 2-231-33.32-c1-0-13
Degree $2$
Conductor $231$
Sign $-0.871 + 0.491i$
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  − 2.45·2-s + (−1.72 − 0.157i)3-s + 4.01·4-s − 3.63i·5-s + (4.22 + 0.385i)6-s + i·7-s − 4.92·8-s + (2.95 + 0.542i)9-s + 8.91i·10-s + (1.88 + 2.72i)11-s + (−6.91 − 0.630i)12-s − 5.76i·13-s − 2.45i·14-s + (−0.571 + 6.27i)15-s + 4.06·16-s − 4.19·17-s + ⋯
L(s)  = 1  − 1.73·2-s + (−0.995 − 0.0907i)3-s + 2.00·4-s − 1.62i·5-s + (1.72 + 0.157i)6-s + 0.377i·7-s − 1.74·8-s + (0.983 + 0.180i)9-s + 2.81i·10-s + (0.568 + 0.822i)11-s + (−1.99 − 0.181i)12-s − 1.59i·13-s − 0.655i·14-s + (−0.147 + 1.61i)15-s + 1.01·16-s − 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.871 + 0.491i$
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: \((2,\ 231,\ (\ :1/2),\ -0.871 + 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0631109 - 0.240449i\)
\(L(\frac12)\) \(\approx\) \(0.0631109 - 0.240449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 + 0.157i)T \)
7 \( 1 - iT \)
11 \( 1 + (-1.88 - 2.72i)T \)
good2 \( 1 + 2.45T + 2T^{2} \)
5 \( 1 + 3.63iT - 5T^{2} \)
13 \( 1 + 5.76iT - 13T^{2} \)
17 \( 1 + 4.19T + 17T^{2} \)
19 \( 1 + 2.05iT - 19T^{2} \)
23 \( 1 + 3.24iT - 23T^{2} \)
29 \( 1 + 4.50T + 29T^{2} \)
31 \( 1 + 6.67T + 31T^{2} \)
37 \( 1 + 0.628T + 37T^{2} \)
41 \( 1 + 1.39T + 41T^{2} \)
43 \( 1 - 0.462iT - 43T^{2} \)
47 \( 1 - 4.50iT - 47T^{2} \)
53 \( 1 + 1.38iT - 53T^{2} \)
59 \( 1 + 0.774iT - 59T^{2} \)
61 \( 1 + 5.90iT - 61T^{2} \)
67 \( 1 + 0.632T + 67T^{2} \)
71 \( 1 + 9.61iT - 71T^{2} \)
73 \( 1 - 14.3iT - 73T^{2} \)
79 \( 1 + 7.56iT - 79T^{2} \)
83 \( 1 + 6.21T + 83T^{2} \)
89 \( 1 + 5.60iT - 89T^{2} \)
97 \( 1 + 8.98T + 97T^{2} \)
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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

−11.56561879457779894447698801820, −10.69131303752134865828063089374, −9.665018839420969711666884118175, −8.981923159547860927773722485106, −8.057437201398881253231244339698, −7.02578901165353961609105269019, −5.74918844814318585038160418830, −4.63836628931253958447949330211, −1.77212826759578289207114419810, −0.39835328165942288516922272698, 1.86067230577550036822184639895, 3.81789405906881411379948312386, 6.10151372382000773255622195073, 6.85488959155611371162833150046, 7.37896920413234444117309007761, 8.950334569009055421105543946436, 9.804829065903859996347619829256, 10.74845606693489174147894501658, 11.23401801670636093689316920779, 11.74719549244757797991032907250

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