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Properties

Label 2-230-5.3-c4-0-38
Degree $2$
Conductor $230$
Sign $-0.995 - 0.0953i$
Analytic cond. $23.7750$
Root an. cond. $4.87597$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 − 2i)2-s + (2.64 + 2.64i)3-s − 8i·4-s + (22.4 − 11.0i)5-s + 10.5·6-s + (−62.5 + 62.5i)7-s + (−16 − 16i)8-s − 66.9i·9-s + (22.7 − 66.9i)10-s − 187.·11-s + (21.1 − 21.1i)12-s + (−63.2 − 63.2i)13-s + 250. i·14-s + (88.6 + 30.1i)15-s − 64·16-s + (−295. + 295. i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.294 + 0.294i)3-s − 0.5i·4-s + (0.896 − 0.442i)5-s + 0.294·6-s + (−1.27 + 1.27i)7-s + (−0.250 − 0.250i)8-s − 0.826i·9-s + (0.227 − 0.669i)10-s − 1.55·11-s + (0.147 − 0.147i)12-s + (−0.374 − 0.374i)13-s + 1.27i·14-s + (0.394 + 0.133i)15-s − 0.250·16-s + (−1.02 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0953i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.995 - 0.0953i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.995 - 0.0953i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6444606478\)
\(L(\frac12)\) \(\approx\) \(0.6444606478\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 + 2i)T \)
5 \( 1 + (-22.4 + 11.0i)T \)
23 \( 1 + (77.9 + 77.9i)T \)
good3 \( 1 + (-2.64 - 2.64i)T + 81iT^{2} \)
7 \( 1 + (62.5 - 62.5i)T - 2.40e3iT^{2} \)
11 \( 1 + 187.T + 1.46e4T^{2} \)
13 \( 1 + (63.2 + 63.2i)T + 2.85e4iT^{2} \)
17 \( 1 + (295. - 295. i)T - 8.35e4iT^{2} \)
19 \( 1 + 514. iT - 1.30e5T^{2} \)
29 \( 1 + 672. iT - 7.07e5T^{2} \)
31 \( 1 + 333.T + 9.23e5T^{2} \)
37 \( 1 + (724. - 724. i)T - 1.87e6iT^{2} \)
41 \( 1 - 556.T + 2.82e6T^{2} \)
43 \( 1 + (-2.22e3 - 2.22e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (1.75e3 - 1.75e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.26e3 + 2.26e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 2.96e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.24e3T + 1.38e7T^{2} \)
67 \( 1 + (5.38e3 - 5.38e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 4.78e3T + 2.54e7T^{2} \)
73 \( 1 + (2.03e3 + 2.03e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 4.85e3iT - 3.89e7T^{2} \)
83 \( 1 + (-4.03e3 - 4.03e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 7.29e3iT - 6.27e7T^{2} \)
97 \( 1 + (-4.62e3 + 4.62e3i)T - 8.85e7iT^{2} \)
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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

−11.00021975620960119248453951713, −9.893806537827166434259479461868, −9.368683534506994450791726590835, −8.474068545103582426036980650408, −6.49789753373657016645018790248, −5.79342368148448028391674070633, −4.69974638124732152458742757887, −3.00019123415867539466798163316, −2.35999534940523581891432954399, −0.15418799809627862970207188228, 2.22287596700516759103443216061, 3.31700443135398800568014469503, 4.84899251769187442641525696507, 5.99186259062641521743491196680, 7.14215453779392779597008519299, 7.56999674848773106770763582698, 9.140643713786025048833068188464, 10.28818292873862727530245190835, 10.73733906223719042297744886299, 12.53247680374296321026060756595

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