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Properties

Label 2-230-5.3-c4-0-18
Degree $2$
Conductor $230$
Sign $-0.605 - 0.796i$
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 + (7.00 + 7.00i)3-s − 8i·4-s + (23.3 + 8.97i)5-s − 28.0·6-s + (−20.2 + 20.2i)7-s + (16 + 16i)8-s + 17.1i·9-s + (−64.6 + 28.7i)10-s + 137.·11-s + (56.0 − 56.0i)12-s + (32.5 + 32.5i)13-s − 80.9i·14-s + (100. + 226. i)15-s − 64·16-s + (−274. + 274. i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.778 + 0.778i)3-s − 0.5i·4-s + (0.933 + 0.358i)5-s − 0.778·6-s + (−0.413 + 0.413i)7-s + (0.250 + 0.250i)8-s + 0.211i·9-s + (−0.646 + 0.287i)10-s + 1.13·11-s + (0.389 − 0.389i)12-s + (0.192 + 0.192i)13-s − 0.413i·14-s + (0.447 + 1.00i)15-s − 0.250·16-s + (−0.948 + 0.948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.796i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.605 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.605 - 0.796i$
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: \((2,\ 230,\ (\ :2),\ -0.605 - 0.796i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.197931864\)
\(L(\frac12)\) \(\approx\) \(2.197931864\)
\(L(3)\) 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 \( 1 + (-23.3 - 8.97i)T \)
23 \( 1 + (-77.9 - 77.9i)T \)
good3 \( 1 + (-7.00 - 7.00i)T + 81iT^{2} \)
7 \( 1 + (20.2 - 20.2i)T - 2.40e3iT^{2} \)
11 \( 1 - 137.T + 1.46e4T^{2} \)
13 \( 1 + (-32.5 - 32.5i)T + 2.85e4iT^{2} \)
17 \( 1 + (274. - 274. i)T - 8.35e4iT^{2} \)
19 \( 1 - 278. iT - 1.30e5T^{2} \)
29 \( 1 + 321. iT - 7.07e5T^{2} \)
31 \( 1 - 1.49e3T + 9.23e5T^{2} \)
37 \( 1 + (1.29e3 - 1.29e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.36e3T + 2.82e6T^{2} \)
43 \( 1 + (850. + 850. i)T + 3.41e6iT^{2} \)
47 \( 1 + (2.77e3 - 2.77e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (3.11e3 + 3.11e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 989. iT - 1.21e7T^{2} \)
61 \( 1 - 3.20e3T + 1.38e7T^{2} \)
67 \( 1 + (4.13e3 - 4.13e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 6.43e3T + 2.54e7T^{2} \)
73 \( 1 + (1.43e3 + 1.43e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 7.00e3iT - 3.89e7T^{2} \)
83 \( 1 + (-2.31e3 - 2.31e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 589. iT - 6.27e7T^{2} \)
97 \( 1 + (-487. + 487. i)T - 8.85e7iT^{2} \)
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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

−11.72104245657155059116284395536, −10.43617595565566259245446927286, −9.722732521627477343931430958219, −9.056760735218608041787324644419, −8.305665390342871570177769324550, −6.59122238663364647798512015831, −6.11231827462796934816156736187, −4.44373942864291746483847952729, −3.13443948296025701760721283521, −1.65420787098406287330324827525, 0.824470338133413993536643711321, 1.97121126706596139646370980022, 3.05936450989063138937804329602, 4.69161965442792677728591805090, 6.47501398577279310529759779631, 7.19302906159022552356771079124, 8.570626779240749978186374629814, 9.112753774391211670702109162392, 10.04264276342545598333012284049, 11.16638370662863114732947137259

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