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Properties

Label 2-230-115.22-c1-0-4
Degree $2$
Conductor $230$
Sign $0.175 - 0.984i$
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  + (0.707 + 0.707i)2-s + (−0.868 + 0.868i)3-s + 1.00i·4-s + (2.22 + 0.185i)5-s − 1.22·6-s + (1.38 − 1.38i)7-s + (−0.707 + 0.707i)8-s + 1.49i·9-s + (1.44 + 1.70i)10-s + 0.642i·11-s + (−0.868 − 0.868i)12-s + (−2.12 + 2.12i)13-s + 1.96·14-s + (−2.09 + 1.77i)15-s − 1.00·16-s + (0.903 − 0.903i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.501 + 0.501i)3-s + 0.500i·4-s + (0.996 + 0.0831i)5-s − 0.501·6-s + (0.525 − 0.525i)7-s + (−0.250 + 0.250i)8-s + 0.497i·9-s + (0.456 + 0.539i)10-s + 0.193i·11-s + (−0.250 − 0.250i)12-s + (−0.589 + 0.589i)13-s + 0.525·14-s + (−0.541 + 0.458i)15-s − 0.250·16-s + (0.219 − 0.219i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20190 + 1.00700i\)
\(L(\frac12)\) \(\approx\) \(1.20190 + 1.00700i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-2.22 - 0.185i)T \)
23 \( 1 + (0.664 + 4.74i)T \)
good3 \( 1 + (0.868 - 0.868i)T - 3iT^{2} \)
7 \( 1 + (-1.38 + 1.38i)T - 7iT^{2} \)
11 \( 1 - 0.642iT - 11T^{2} \)
13 \( 1 + (2.12 - 2.12i)T - 13iT^{2} \)
17 \( 1 + (-0.903 + 0.903i)T - 17iT^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
29 \( 1 - 0.214iT - 29T^{2} \)
31 \( 1 - 6.15T + 31T^{2} \)
37 \( 1 + (-7.20 + 7.20i)T - 37iT^{2} \)
41 \( 1 - 3.33T + 41T^{2} \)
43 \( 1 + (7.85 + 7.85i)T + 43iT^{2} \)
47 \( 1 + (4.15 + 4.15i)T + 47iT^{2} \)
53 \( 1 + (2.93 + 2.93i)T + 53iT^{2} \)
59 \( 1 + 7.08iT - 59T^{2} \)
61 \( 1 - 5.29iT - 61T^{2} \)
67 \( 1 + (3.65 - 3.65i)T - 67iT^{2} \)
71 \( 1 + 1.83T + 71T^{2} \)
73 \( 1 + (5.20 - 5.20i)T - 73iT^{2} \)
79 \( 1 - 6.19T + 79T^{2} \)
83 \( 1 + (-6.04 - 6.04i)T + 83iT^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (-10.7 + 10.7i)T - 97iT^{2} \)
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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

−12.54342718536250465452046174649, −11.40143190463261600003323685868, −10.49711862422024494892201726159, −9.700627293877878205541999036522, −8.390590169988212114330682555908, −7.19261445498779913124089368493, −6.15495630195151136016314606611, −5.05845451486696573122624407635, −4.33358509160324092712556202016, −2.29969711745704398429089627287, 1.45426937586484550235346958459, 2.91619472931517622440618608786, 4.77602732028626855617807984035, 5.79229136511809457073048516227, 6.48538950769411380043401525190, 8.026384984914898396401582682483, 9.338908348964128807899121829482, 10.13919657912392255404142975466, 11.29919797668726866689124510334, 12.04446532825836263810933645825

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