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Properties

Label 2-231-231.65-c1-0-2
Degree $2$
Conductor $231$
Sign $-0.976 + 0.215i$
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  + (−0.939 + 1.62i)2-s + (−1.69 + 0.370i)3-s + (−0.765 − 1.32i)4-s + (2.63 + 1.52i)5-s + (0.986 − 3.10i)6-s + (−0.738 + 2.54i)7-s − 0.882·8-s + (2.72 − 1.25i)9-s + (−4.95 + 2.86i)10-s + (−0.988 + 3.16i)11-s + (1.78 + 1.95i)12-s + 1.24i·13-s + (−3.44 − 3.58i)14-s + (−5.02 − 1.59i)15-s + (2.35 − 4.08i)16-s + (−2.83 − 4.90i)17-s + ⋯
L(s)  = 1  + (−0.664 + 1.15i)2-s + (−0.976 + 0.213i)3-s + (−0.382 − 0.662i)4-s + (1.17 + 0.680i)5-s + (0.402 − 1.26i)6-s + (−0.279 + 0.960i)7-s − 0.311·8-s + (0.908 − 0.417i)9-s + (−1.56 + 0.904i)10-s + (−0.298 + 0.954i)11-s + (0.515 + 0.565i)12-s + 0.345i·13-s + (−0.919 − 0.959i)14-s + (−1.29 − 0.412i)15-s + (0.589 − 1.02i)16-s + (−0.687 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $-0.976 + 0.215i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ -0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0672639 - 0.617766i\)
\(L(\frac12)\) \(\approx\) \(0.0672639 - 0.617766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 - 0.370i)T \)
7 \( 1 + (0.738 - 2.54i)T \)
11 \( 1 + (0.988 - 3.16i)T \)
good2 \( 1 + (0.939 - 1.62i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-2.63 - 1.52i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.24iT - 13T^{2} \)
17 \( 1 + (2.83 + 4.90i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.532 - 0.307i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.41 + 2.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.47T + 29T^{2} \)
31 \( 1 + (-3.18 - 5.50i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.15 - 5.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 2.31iT - 43T^{2} \)
47 \( 1 + (-9.35 - 5.40i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.58 - 1.48i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.38 - 3.10i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.24 - 4.76i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.31 - 4.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.90iT - 71T^{2} \)
73 \( 1 + (1.01 - 0.586i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.34 + 1.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 + (1.19 + 0.690i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.19T + 97T^{2} \)
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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

−12.53601709325633013787142650867, −11.78399004324334041865464337267, −10.43709562138801420427256029023, −9.659439327924997326414972931216, −9.007945188011107031352010385507, −7.39653467330303161203610908294, −6.54345007718483434406600471166, −5.94051758181333979893535818341, −4.91538984944174103580060665583, −2.47030764837357287949492230481, 0.70241124930581318358082762859, 2.00142443363869974331979848026, 3.95628448671511876034594436211, 5.62916495227723140553415741984, 6.27123246810500616260283325214, 7.927109674604410888784468547655, 9.188054857452143411826894416523, 10.05414675023852825531468440220, 10.67274627961546695932469120645, 11.38217068266481084275338964232

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