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Properties

Label 2-230-5.3-c4-0-35
Degree $2$
Conductor $230$
Sign $-0.720 + 0.693i$
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 + (−0.310 − 0.310i)3-s − 8i·4-s + (6.21 − 24.2i)5-s − 1.24·6-s + (40.7 − 40.7i)7-s + (−16 − 16i)8-s − 80.8i·9-s + (−36.0 − 60.8i)10-s + 96.9·11-s + (−2.48 + 2.48i)12-s + (160. + 160. i)13-s − 163. i·14-s + (−9.44 + 5.58i)15-s − 64·16-s + (−324. + 324. i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.0344 − 0.0344i)3-s − 0.5i·4-s + (0.248 − 0.968i)5-s − 0.0344·6-s + (0.832 − 0.832i)7-s + (−0.250 − 0.250i)8-s − 0.997i·9-s + (−0.360 − 0.608i)10-s + 0.801·11-s + (−0.0172 + 0.0172i)12-s + (0.947 + 0.947i)13-s − 0.832i·14-s + (−0.0419 + 0.0248i)15-s − 0.250·16-s + (−1.12 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.720 + 0.693i$
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: \((2,\ 230,\ (\ :2),\ -0.720 + 0.693i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.792899054\)
\(L(\frac12)\) \(\approx\) \(2.792899054\)
\(L(3)\) 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 \( 1 + (-6.21 + 24.2i)T \)
23 \( 1 + (77.9 + 77.9i)T \)
good3 \( 1 + (0.310 + 0.310i)T + 81iT^{2} \)
7 \( 1 + (-40.7 + 40.7i)T - 2.40e3iT^{2} \)
11 \( 1 - 96.9T + 1.46e4T^{2} \)
13 \( 1 + (-160. - 160. i)T + 2.85e4iT^{2} \)
17 \( 1 + (324. - 324. i)T - 8.35e4iT^{2} \)
19 \( 1 + 288. iT - 1.30e5T^{2} \)
29 \( 1 + 458. iT - 7.07e5T^{2} \)
31 \( 1 - 417.T + 9.23e5T^{2} \)
37 \( 1 + (-994. + 994. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.74e3T + 2.82e6T^{2} \)
43 \( 1 + (1.50e3 + 1.50e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-659. + 659. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-2.39e3 - 2.39e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 4.44e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.01e3T + 1.38e7T^{2} \)
67 \( 1 + (4.84e3 - 4.84e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 2.19e3T + 2.54e7T^{2} \)
73 \( 1 + (1.48e3 + 1.48e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 2.69e3iT - 3.89e7T^{2} \)
83 \( 1 + (-864. - 864. i)T + 4.74e7iT^{2} \)
89 \( 1 - 6.62e3iT - 6.27e7T^{2} \)
97 \( 1 + (2.51e3 - 2.51e3i)T - 8.85e7iT^{2} \)
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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

−11.44206360434982880794049831970, −10.38342625760846375420969737019, −9.114205525533606451346099836308, −8.579451480012955264620553811591, −6.86722662789110611978240162342, −5.95049417626844713217965825466, −4.37698517491992142573140933755, −4.00633262274955114736514725687, −1.79254810259615831033513451757, −0.841399473718781280762317970986, 1.96150434977672064356979558763, 3.22478448289942078298765867334, 4.75672451137116985474482579900, 5.75135116665376725830235300081, 6.71902082371349304842704160952, 7.891143602172530622935126159199, 8.687739796786494960387870630074, 10.08939481161550809807156795024, 11.23153371613206458534808251434, 11.67285391479792802614467774441

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