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Properties

Label 2-230-5.3-c4-0-42
Degree $2$
Conductor $230$
Sign $0.0336 - 0.999i$
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 + (−4.66 − 4.66i)3-s − 8i·4-s + (−12.4 − 21.6i)5-s − 18.6·6-s + (−24.9 + 24.9i)7-s + (−16 − 16i)8-s − 37.4i·9-s + (−68.2 − 18.5i)10-s + 42.7·11-s + (−37.3 + 37.3i)12-s + (−124. − 124. i)13-s + 99.9i·14-s + (−43.2 + 159. i)15-s − 64·16-s + (−82.2 + 82.2i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.518 − 0.518i)3-s − 0.5i·4-s + (−0.496 − 0.867i)5-s − 0.518·6-s + (−0.510 + 0.510i)7-s + (−0.250 − 0.250i)8-s − 0.462i·9-s + (−0.682 − 0.185i)10-s + 0.353·11-s + (−0.259 + 0.259i)12-s + (−0.735 − 0.735i)13-s + 0.510i·14-s + (−0.192 + 0.707i)15-s − 0.250·16-s + (−0.284 + 0.284i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0336 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0336 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.0336 - 0.999i$
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: \((2,\ 230,\ (\ :2),\ 0.0336 - 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.2169213041\)
\(L(\frac12)\) \(\approx\) \(0.2169213041\)
\(L(3)\) 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 \( 1 + (12.4 + 21.6i)T \)
23 \( 1 + (-77.9 - 77.9i)T \)
good3 \( 1 + (4.66 + 4.66i)T + 81iT^{2} \)
7 \( 1 + (24.9 - 24.9i)T - 2.40e3iT^{2} \)
11 \( 1 - 42.7T + 1.46e4T^{2} \)
13 \( 1 + (124. + 124. i)T + 2.85e4iT^{2} \)
17 \( 1 + (82.2 - 82.2i)T - 8.35e4iT^{2} \)
19 \( 1 - 142. iT - 1.30e5T^{2} \)
29 \( 1 - 448. iT - 7.07e5T^{2} \)
31 \( 1 - 1.20e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.16e3 + 1.16e3i)T - 1.87e6iT^{2} \)
41 \( 1 + 983.T + 2.82e6T^{2} \)
43 \( 1 + (-1.26e3 - 1.26e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (1.74e3 - 1.74e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.00e3 + 2.00e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 957. iT - 1.21e7T^{2} \)
61 \( 1 + 3.39e3T + 1.38e7T^{2} \)
67 \( 1 + (3.86e3 - 3.86e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 535.T + 2.54e7T^{2} \)
73 \( 1 + (-2.47e3 - 2.47e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 4.13e3iT - 3.89e7T^{2} \)
83 \( 1 + (7.57e3 + 7.57e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 8.59e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.26e4 - 1.26e4i)T - 8.85e7iT^{2} \)
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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

−11.20735346002269696687008690686, −9.851727306546852844016264669635, −9.008386570126309378800627558256, −7.75205246244864053694506886219, −6.44062569136017374650759477013, −5.55496829142939742623731874821, −4.38295635517504014075315002152, −3.05071725807496911451002971765, −1.30390734482292510702467224091, −0.06928403202675558787336546962, 2.65934558215902146758393739627, 4.03573539983027833109363459047, 4.84476166393960424918369949671, 6.31900587120544613425360162077, 7.01980117728544749415569645233, 8.032058752845464931407245967640, 9.547951530817085346461219277733, 10.44675144829200602802574478117, 11.40074932181730355071644086853, 12.06499447438983763410687086448

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