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Properties

Label 2-230-5.3-c4-0-8
Degree $2$
Conductor $230$
Sign $0.845 - 0.534i$
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 + (−10.5 − 10.5i)3-s − 8i·4-s + (10.9 − 22.4i)5-s + 42.0·6-s + (−20.3 + 20.3i)7-s + (16 + 16i)8-s + 139. i·9-s + (23.0 + 66.8i)10-s + 3.75·11-s + (−84.0 + 84.0i)12-s + (−110. − 110. i)13-s − 81.4i·14-s + (−350. + 121. i)15-s − 64·16-s + (−262. + 262. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−1.16 − 1.16i)3-s − 0.5i·4-s + (0.437 − 0.899i)5-s + 1.16·6-s + (−0.415 + 0.415i)7-s + (0.250 + 0.250i)8-s + 1.72i·9-s + (0.230 + 0.668i)10-s + 0.0310·11-s + (−0.583 + 0.583i)12-s + (−0.656 − 0.656i)13-s − 0.415i·14-s + (−1.55 + 0.538i)15-s − 0.250·16-s + (−0.908 + 0.908i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.845 - 0.534i$
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: \((2,\ 230,\ (\ :2),\ 0.845 - 0.534i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5219491471\)
\(L(\frac12)\) \(\approx\) \(0.5219491471\)
\(L(3)\) 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 \( 1 + (-10.9 + 22.4i)T \)
23 \( 1 + (77.9 + 77.9i)T \)
good3 \( 1 + (10.5 + 10.5i)T + 81iT^{2} \)
7 \( 1 + (20.3 - 20.3i)T - 2.40e3iT^{2} \)
11 \( 1 - 3.75T + 1.46e4T^{2} \)
13 \( 1 + (110. + 110. i)T + 2.85e4iT^{2} \)
17 \( 1 + (262. - 262. i)T - 8.35e4iT^{2} \)
19 \( 1 - 305. iT - 1.30e5T^{2} \)
29 \( 1 - 27.2iT - 7.07e5T^{2} \)
31 \( 1 - 843.T + 9.23e5T^{2} \)
37 \( 1 + (-323. + 323. i)T - 1.87e6iT^{2} \)
41 \( 1 - 105.T + 2.82e6T^{2} \)
43 \( 1 + (-1.72e3 - 1.72e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (574. - 574. i)T - 4.87e6iT^{2} \)
53 \( 1 + (790. + 790. i)T + 7.89e6iT^{2} \)
59 \( 1 + 194. iT - 1.21e7T^{2} \)
61 \( 1 - 2.98e3T + 1.38e7T^{2} \)
67 \( 1 + (-1.25e3 + 1.25e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 5.43e3T + 2.54e7T^{2} \)
73 \( 1 + (6.10e3 + 6.10e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 2.58e3iT - 3.89e7T^{2} \)
83 \( 1 + (580. + 580. i)T + 4.74e7iT^{2} \)
89 \( 1 + 1.89e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.05e4 + 1.05e4i)T - 8.85e7iT^{2} \)
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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

−11.81451019581894630884706074997, −10.65691586636813256423297594285, −9.653669619951000915637075946347, −8.459549960457913581322999088332, −7.59654466115043281908852559373, −6.30513816644833162512590214688, −5.88987686527841849529074096055, −4.76486511081306672240694100157, −2.07082340884421633849752579258, −0.846997815357481007142275868548, 0.33351602349902465750074068715, 2.55724933562467966730877865330, 3.96575873068517454447427618096, 5.03103186037199760598675913258, 6.42131321449768481210423295846, 7.17018915389396446170257694130, 9.102328814386601083377412667097, 9.820399890602482644517302226468, 10.44740700991764243205023017152, 11.31177094535085443102586114921

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