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Properties

Label 2-230-5.3-c4-0-5
Degree $2$
Conductor $230$
Sign $-0.914 - 0.404i$
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 + (11.7 + 11.7i)3-s − 8i·4-s + (−24.7 + 3.42i)5-s + 47.0·6-s + (−22.8 + 22.8i)7-s + (−16 − 16i)8-s + 195. i·9-s + (−42.6 + 56.3i)10-s − 90.9·11-s + (94.1 − 94.1i)12-s + (−139. − 139. i)13-s + 91.4i·14-s + (−331. − 250. i)15-s − 64·16-s + (−293. + 293. i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (1.30 + 1.30i)3-s − 0.5i·4-s + (−0.990 + 0.137i)5-s + 1.30·6-s + (−0.466 + 0.466i)7-s + (−0.250 − 0.250i)8-s + 2.41i·9-s + (−0.426 + 0.563i)10-s − 0.751·11-s + (0.653 − 0.653i)12-s + (−0.827 − 0.827i)13-s + 0.466i·14-s + (−1.47 − 1.11i)15-s − 0.250·16-s + (−1.01 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.914 - 0.404i$
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: \((2,\ 230,\ (\ :2),\ -0.914 - 0.404i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.357522762\)
\(L(\frac12)\) \(\approx\) \(1.357522762\)
\(L(3)\) 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 \( 1 + (24.7 - 3.42i)T \)
23 \( 1 + (-77.9 - 77.9i)T \)
good3 \( 1 + (-11.7 - 11.7i)T + 81iT^{2} \)
7 \( 1 + (22.8 - 22.8i)T - 2.40e3iT^{2} \)
11 \( 1 + 90.9T + 1.46e4T^{2} \)
13 \( 1 + (139. + 139. i)T + 2.85e4iT^{2} \)
17 \( 1 + (293. - 293. i)T - 8.35e4iT^{2} \)
19 \( 1 + 533. iT - 1.30e5T^{2} \)
29 \( 1 - 1.36e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.64e3T + 9.23e5T^{2} \)
37 \( 1 + (1.07e3 - 1.07e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.77e3T + 2.82e6T^{2} \)
43 \( 1 + (1.38e3 + 1.38e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.77e3 + 1.77e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-3.87e3 - 3.87e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 3.62e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.93e3T + 1.38e7T^{2} \)
67 \( 1 + (1.95e3 - 1.95e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.40e3T + 2.54e7T^{2} \)
73 \( 1 + (1.37e3 + 1.37e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 9.22e3iT - 3.89e7T^{2} \)
83 \( 1 + (567. + 567. i)T + 4.74e7iT^{2} \)
89 \( 1 + 62.0iT - 6.27e7T^{2} \)
97 \( 1 + (-5.61e3 + 5.61e3i)T - 8.85e7iT^{2} \)
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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

−11.95958672812638638458820787307, −10.60760214649578471268671698351, −10.33919828683597735304832592463, −8.971039458928589094949683665130, −8.428145331458668345253348724540, −7.11950002972372765371158876505, −5.14037911652989342265353918217, −4.37527776348569262526018944431, −3.15452358761859206753744807639, −2.62592563809827931583563519248, 0.30062739144029735647435588543, 2.26096417823701369113789913798, 3.37889825755747791528210436570, 4.52124442128642110173623087116, 6.46610040857010256085538694062, 7.22935956757772785334719768456, 7.894145589781361391838693813929, 8.676464630549820782600402445222, 9.877359227301430974933520522507, 11.74391349689188853313599152571

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