Properties

Label 2-230-115.57-c1-0-7
Degree $2$
Conductor $230$
Sign $-0.264 + 0.964i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.977 − 0.212i)2-s + (−1.53 − 2.04i)3-s + (0.909 + 0.415i)4-s + (1.63 − 1.52i)5-s + (1.06 + 2.32i)6-s + (5.04 + 0.361i)7-s + (−0.800 − 0.599i)8-s + (−0.993 + 3.38i)9-s + (−1.92 + 1.13i)10-s + (−1.44 + 2.24i)11-s + (−0.543 − 2.49i)12-s + (−0.433 − 6.05i)13-s + (−4.85 − 1.42i)14-s + (−5.62 − 1.02i)15-s + (0.654 + 0.755i)16-s + (−1.40 − 0.522i)17-s + ⋯
L(s)  = 1  + (−0.690 − 0.150i)2-s + (−0.884 − 1.18i)3-s + (0.454 + 0.207i)4-s + (0.733 − 0.680i)5-s + (0.433 + 0.948i)6-s + (1.90 + 0.136i)7-s + (−0.283 − 0.211i)8-s + (−0.331 + 1.12i)9-s + (−0.608 + 0.359i)10-s + (−0.435 + 0.678i)11-s + (−0.156 − 0.720i)12-s + (−0.120 − 1.68i)13-s + (−1.29 − 0.381i)14-s + (−1.45 − 0.264i)15-s + (0.163 + 0.188i)16-s + (−0.339 − 0.126i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.264 + 0.964i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.264 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.526054 - 0.690014i\)
\(L(\frac12)\) \(\approx\) \(0.526054 - 0.690014i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.977 + 0.212i)T \)
5 \( 1 + (-1.63 + 1.52i)T \)
23 \( 1 + (-1.38 - 4.59i)T \)
good3 \( 1 + (1.53 + 2.04i)T + (-0.845 + 2.87i)T^{2} \)
7 \( 1 + (-5.04 - 0.361i)T + (6.92 + 0.996i)T^{2} \)
11 \( 1 + (1.44 - 2.24i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (0.433 + 6.05i)T + (-12.8 + 1.85i)T^{2} \)
17 \( 1 + (1.40 + 0.522i)T + (12.8 + 11.1i)T^{2} \)
19 \( 1 + (-0.914 + 2.00i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (5.74 - 2.62i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (-0.449 - 3.12i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (2.64 - 1.44i)T + (20.0 - 31.1i)T^{2} \)
41 \( 1 + (-3.21 + 0.943i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-0.0872 + 0.0653i)T + (12.1 - 41.2i)T^{2} \)
47 \( 1 + (1.68 + 1.68i)T + 47iT^{2} \)
53 \( 1 + (-0.717 + 10.0i)T + (-52.4 - 7.54i)T^{2} \)
59 \( 1 + (-9.39 - 8.13i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (2.07 - 0.298i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (0.829 - 3.81i)T + (-60.9 - 27.8i)T^{2} \)
71 \( 1 + (3.32 - 2.13i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-2.82 - 7.56i)T + (-55.1 + 47.8i)T^{2} \)
79 \( 1 + (0.225 - 0.260i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-1.21 - 2.21i)T + (-44.8 + 69.8i)T^{2} \)
89 \( 1 + (-1.49 + 10.4i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (2.07 - 3.80i)T + (-52.4 - 81.6i)T^{2} \)
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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

−11.83243687254167751722850925353, −11.10548433713531440388169396113, −10.17314242844575205223968651798, −8.772351834845997348892155522742, −7.86002430113218040365320790732, −7.17787106718910262664401226203, −5.56574505496092337788427587272, −5.10453515656264470022561768513, −2.12795765713702332195925598088, −1.10660699141074088891113359299, 1.99086066396579932071093343683, 4.24535699738198719660441957126, 5.27022248379697087346796210716, 6.24625723785572426123492343907, 7.56631321499783250399081819045, 8.799041706593360076364010458618, 9.713541642989591482481131878083, 10.83246848436756630560621376873, 11.03073439195797579805015344553, 11.81502657398353163517069204177

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