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Properties

Label 2-230-5.2-c4-0-15
Degree $2$
Conductor $230$
Sign $-0.478 - 0.878i$
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 + (7.61 − 7.61i)3-s + 8i·4-s + (−1.36 + 24.9i)5-s + 30.4·6-s + (45.0 + 45.0i)7-s + (−16 + 16i)8-s − 34.8i·9-s + (−52.6 + 47.1i)10-s − 201.·11-s + (60.8 + 60.8i)12-s + (−180. + 180. i)13-s + 180. i·14-s + (179. + 200. i)15-s − 64·16-s + (66.9 + 66.9i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.845 − 0.845i)3-s + 0.5i·4-s + (−0.0547 + 0.998i)5-s + 0.845·6-s + (0.920 + 0.920i)7-s + (−0.250 + 0.250i)8-s − 0.430i·9-s + (−0.526 + 0.471i)10-s − 1.66·11-s + (0.422 + 0.422i)12-s + (−1.06 + 1.06i)13-s + 0.920i·14-s + (0.798 + 0.890i)15-s − 0.250·16-s + (0.231 + 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.478 - 0.878i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.478 - 0.878i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.726584417\)
\(L(\frac12)\) \(\approx\) \(2.726584417\)
\(L(3)\) 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 \( 1 + (1.36 - 24.9i)T \)
23 \( 1 + (77.9 - 77.9i)T \)
good3 \( 1 + (-7.61 + 7.61i)T - 81iT^{2} \)
7 \( 1 + (-45.0 - 45.0i)T + 2.40e3iT^{2} \)
11 \( 1 + 201.T + 1.46e4T^{2} \)
13 \( 1 + (180. - 180. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-66.9 - 66.9i)T + 8.35e4iT^{2} \)
19 \( 1 + 264. iT - 1.30e5T^{2} \)
29 \( 1 + 1.18e3iT - 7.07e5T^{2} \)
31 \( 1 - 123.T + 9.23e5T^{2} \)
37 \( 1 + (-1.62e3 - 1.62e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.57e3T + 2.82e6T^{2} \)
43 \( 1 + (-237. + 237. i)T - 3.41e6iT^{2} \)
47 \( 1 + (-375. - 375. i)T + 4.87e6iT^{2} \)
53 \( 1 + (781. - 781. i)T - 7.89e6iT^{2} \)
59 \( 1 - 6.25e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.72e3T + 1.38e7T^{2} \)
67 \( 1 + (-4.26e3 - 4.26e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 2.27e3T + 2.54e7T^{2} \)
73 \( 1 + (1.76e3 - 1.76e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 3.54e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.01e3 - 1.01e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 1.03e4iT - 6.27e7T^{2} \)
97 \( 1 + (-3.95e3 - 3.95e3i)T + 8.85e7iT^{2} \)
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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

−11.97856829520339600516172939855, −11.18682013548028274118686119484, −9.813338682420004531967549485922, −8.431999297271468109866300394132, −7.75290311353410309673757209486, −7.08165920692416391587045403179, −5.77149741431840291076780424837, −4.59310242048411827342300622148, −2.61916801218112389397132991259, −2.33054702190928745427597547809, 0.66063936180232635292825645621, 2.41246208614762136233020723700, 3.70186407884232346541328114782, 4.78892120570283948698089559661, 5.34801782795077998598832074116, 7.67358646120641866108318661180, 8.196463976683239450013612018457, 9.533783390681605905026344384124, 10.25356771185020365448645998240, 11.01626206735909248746487010832

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