Properties

Label 2-230-5.3-c4-0-30
Degree $2$
Conductor $230$
Sign $0.997 + 0.0738i$
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 + (6.39 + 6.39i)3-s − 8i·4-s + (22.1 − 11.5i)5-s − 25.5·6-s + (18.6 − 18.6i)7-s + (16 + 16i)8-s + 0.837i·9-s + (−21.2 + 67.4i)10-s + 60.8·11-s + (51.1 − 51.1i)12-s + (−52.7 − 52.7i)13-s + 74.7i·14-s + (215. + 68.0i)15-s − 64·16-s + (36.5 − 36.5i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.710 + 0.710i)3-s − 0.5i·4-s + (0.887 − 0.461i)5-s − 0.710·6-s + (0.381 − 0.381i)7-s + (0.250 + 0.250i)8-s + 0.0103i·9-s + (−0.212 + 0.674i)10-s + 0.503·11-s + (0.355 − 0.355i)12-s + (−0.312 − 0.312i)13-s + 0.381i·14-s + (0.958 + 0.302i)15-s − 0.250·16-s + (0.126 − 0.126i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0738i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.997 + 0.0738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.997 + 0.0738i$
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: \((2,\ 230,\ (\ :2),\ 0.997 + 0.0738i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.275367167\)
\(L(\frac12)\) \(\approx\) \(2.275367167\)
\(L(3)\) 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 \( 1 + (-22.1 + 11.5i)T \)
23 \( 1 + (77.9 + 77.9i)T \)
good3 \( 1 + (-6.39 - 6.39i)T + 81iT^{2} \)
7 \( 1 + (-18.6 + 18.6i)T - 2.40e3iT^{2} \)
11 \( 1 - 60.8T + 1.46e4T^{2} \)
13 \( 1 + (52.7 + 52.7i)T + 2.85e4iT^{2} \)
17 \( 1 + (-36.5 + 36.5i)T - 8.35e4iT^{2} \)
19 \( 1 + 628. iT - 1.30e5T^{2} \)
29 \( 1 + 209. iT - 7.07e5T^{2} \)
31 \( 1 + 1.45e3T + 9.23e5T^{2} \)
37 \( 1 + (-810. + 810. i)T - 1.87e6iT^{2} \)
41 \( 1 + 10.7T + 2.82e6T^{2} \)
43 \( 1 + (-931. - 931. i)T + 3.41e6iT^{2} \)
47 \( 1 + (487. - 487. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-1.52e3 - 1.52e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 5.35e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.08e3T + 1.38e7T^{2} \)
67 \( 1 + (-4.05e3 + 4.05e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 1.06e3T + 2.54e7T^{2} \)
73 \( 1 + (1.81e3 + 1.81e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 6.10e3iT - 3.89e7T^{2} \)
83 \( 1 + (-7.07e3 - 7.07e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 3.66e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.70e3 - 1.70e3i)T - 8.85e7iT^{2} \)
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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

−11.20954865615699557891956730760, −10.22888149886155353603147585669, −9.255487886310247888092823311256, −8.983510710292787618212845044583, −7.69394979565958194045843021308, −6.52392567617682945287078592179, −5.25920637274453538593985384585, −4.19857295917859504429773079793, −2.51620402029249454863685741495, −0.874412686949120004758157720738, 1.57113778953199978610635191921, 2.19966788565504360784792569870, 3.56758960227579270894213725443, 5.44006490946078576587718419744, 6.73479024431920674412786063933, 7.74695352646464411320078126779, 8.640636399146519514353539817715, 9.588015469061556485748118147617, 10.46543673253119559877357112633, 11.54775505376571223568048517338

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