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Properties

Label 2-230-5.2-c4-0-12
Degree $2$
Conductor $230$
Sign $0.999 - 0.0247i$
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 + (6.64 − 6.64i)3-s + 8i·4-s + (−21.5 − 12.6i)5-s − 26.5·6-s + (27.2 + 27.2i)7-s + (16 − 16i)8-s − 7.31i·9-s + (17.9 + 68.3i)10-s + 37.9·11-s + (53.1 + 53.1i)12-s + (−142. + 142. i)13-s − 108. i·14-s + (−227. + 59.6i)15-s − 64·16-s + (245. + 245. i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.738 − 0.738i)3-s + 0.5i·4-s + (−0.863 − 0.504i)5-s − 0.738·6-s + (0.555 + 0.555i)7-s + (0.250 − 0.250i)8-s − 0.0903i·9-s + (0.179 + 0.683i)10-s + 0.313·11-s + (0.369 + 0.369i)12-s + (−0.844 + 0.844i)13-s − 0.555i·14-s + (−1.01 + 0.264i)15-s − 0.250·16-s + (0.848 + 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0247i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.999 - 0.0247i$
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: \((2,\ 230,\ (\ :2),\ 0.999 - 0.0247i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.505318605\)
\(L(\frac12)\) \(\approx\) \(1.505318605\)
\(L(3)\) 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 \( 1 + (21.5 + 12.6i)T \)
23 \( 1 + (-77.9 + 77.9i)T \)
good3 \( 1 + (-6.64 + 6.64i)T - 81iT^{2} \)
7 \( 1 + (-27.2 - 27.2i)T + 2.40e3iT^{2} \)
11 \( 1 - 37.9T + 1.46e4T^{2} \)
13 \( 1 + (142. - 142. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-245. - 245. i)T + 8.35e4iT^{2} \)
19 \( 1 - 52.3iT - 1.30e5T^{2} \)
29 \( 1 - 529. iT - 7.07e5T^{2} \)
31 \( 1 - 291.T + 9.23e5T^{2} \)
37 \( 1 + (1.11e3 + 1.11e3i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.60e3T + 2.82e6T^{2} \)
43 \( 1 + (1.52e3 - 1.52e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-358. - 358. i)T + 4.87e6iT^{2} \)
53 \( 1 + (-2.46e3 + 2.46e3i)T - 7.89e6iT^{2} \)
59 \( 1 - 6.45e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.44e3T + 1.38e7T^{2} \)
67 \( 1 + (150. + 150. i)T + 2.01e7iT^{2} \)
71 \( 1 - 4.53e3T + 2.54e7T^{2} \)
73 \( 1 + (-4.99e3 + 4.99e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 4.22e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.21e3 - 1.21e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.11e4iT - 6.27e7T^{2} \)
97 \( 1 + (-3.43e3 - 3.43e3i)T + 8.85e7iT^{2} \)
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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

−11.80900389478572621707186079468, −10.66616202309815754855925209370, −9.311228631643701348676308016143, −8.503316351228898651647690004702, −7.86742530304672702151959663905, −6.96433762187218849304260429522, −5.09391375570417509871839303583, −3.76568455219271914762738896555, −2.32529666127561462285534166529, −1.25085195190203549138585090134, 0.62247938313811028831357878665, 2.85563932128340950549581115731, 4.00374437249791277536695945127, 5.14299294565476855784589498002, 6.84918829069673649127858695832, 7.71599845886926913593338547139, 8.432668853594026474814376357504, 9.645358278839698173785674184965, 10.27504679162006633525186940788, 11.33478013501108131332606461851

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