Properties

Label 2-230-5.3-c4-0-2
Degree $2$
Conductor $230$
Sign $-0.710 - 0.703i$
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 + (−6.81 − 6.81i)3-s − 8i·4-s + (24.3 + 5.62i)5-s + 27.2·6-s + (−23.8 + 23.8i)7-s + (16 + 16i)8-s + 11.9i·9-s + (−59.9 + 37.4i)10-s − 79.3·11-s + (−54.5 + 54.5i)12-s + (−18.8 − 18.8i)13-s − 95.3i·14-s + (−127. − 204. i)15-s − 64·16-s + (178. − 178. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.757 − 0.757i)3-s − 0.5i·4-s + (0.974 + 0.224i)5-s + 0.757·6-s + (−0.486 + 0.486i)7-s + (0.250 + 0.250i)8-s + 0.147i·9-s + (−0.599 + 0.374i)10-s − 0.655·11-s + (−0.378 + 0.378i)12-s + (−0.111 − 0.111i)13-s − 0.486i·14-s + (−0.567 − 0.908i)15-s − 0.250·16-s + (0.616 − 0.616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.710 - 0.703i$
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: \((2,\ 230,\ (\ :2),\ -0.710 - 0.703i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.3775596990\)
\(L(\frac12)\) \(\approx\) \(0.3775596990\)
\(L(3)\) 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 \( 1 + (-24.3 - 5.62i)T \)
23 \( 1 + (-77.9 - 77.9i)T \)
good3 \( 1 + (6.81 + 6.81i)T + 81iT^{2} \)
7 \( 1 + (23.8 - 23.8i)T - 2.40e3iT^{2} \)
11 \( 1 + 79.3T + 1.46e4T^{2} \)
13 \( 1 + (18.8 + 18.8i)T + 2.85e4iT^{2} \)
17 \( 1 + (-178. + 178. i)T - 8.35e4iT^{2} \)
19 \( 1 + 53.2iT - 1.30e5T^{2} \)
29 \( 1 + 491. iT - 7.07e5T^{2} \)
31 \( 1 + 739.T + 9.23e5T^{2} \)
37 \( 1 + (830. - 830. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.96e3T + 2.82e6T^{2} \)
43 \( 1 + (-1.35e3 - 1.35e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (2.26e3 - 2.26e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-1.90e3 - 1.90e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 1.01e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.14e3T + 1.38e7T^{2} \)
67 \( 1 + (1.46e3 - 1.46e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 6.52e3T + 2.54e7T^{2} \)
73 \( 1 + (-2.15e3 - 2.15e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 6.66e3iT - 3.89e7T^{2} \)
83 \( 1 + (-1.68e3 - 1.68e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 3.80e3iT - 6.27e7T^{2} \)
97 \( 1 + (548. - 548. i)T - 8.85e7iT^{2} \)
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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

−11.94333143204311911605410986693, −10.84816473581377507851866042863, −9.848173071345426030871973346116, −9.097090460153469373290724095697, −7.69919948948026970529281735642, −6.75086226858562696030534554109, −5.94156915118193790410685779409, −5.21466140883037171829240936487, −2.79837071770145170071732399376, −1.30872313057049731425427201295, 0.17092465953803871280801879327, 1.87253168323410813994483455395, 3.49922591405965164564809033192, 4.91489115400952077156270001895, 5.81592344856039217033087437412, 7.12822397993732499942195920909, 8.507952022362070050585405987727, 9.614236079407990520751517035359, 10.35869134083153763652103196136, 10.71337043806152455693338809548

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