Properties

Label 2-230-115.22-c3-0-24
Degree $2$
Conductor $230$
Sign $0.845 + 0.534i$
Analytic cond. $13.5704$
Root an. cond. $3.68380$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 1.41i)2-s + (0.422 − 0.422i)3-s + 4.00i·4-s + (11.1 − 0.923i)5-s − 1.19·6-s + (23.7 − 23.7i)7-s + (5.65 − 5.65i)8-s + 26.6i·9-s + (−17.0 − 14.4i)10-s + 53.9i·11-s + (1.69 + 1.69i)12-s + (33.6 − 33.6i)13-s − 67.2·14-s + (4.31 − 5.09i)15-s − 16.0·16-s + (−38.4 + 38.4i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.0813 − 0.0813i)3-s + 0.500i·4-s + (0.996 − 0.0825i)5-s − 0.0813·6-s + (1.28 − 1.28i)7-s + (0.250 − 0.250i)8-s + 0.986i·9-s + (−0.539 − 0.456i)10-s + 1.47i·11-s + (0.0406 + 0.0406i)12-s + (0.717 − 0.717i)13-s − 1.28·14-s + (0.0743 − 0.0877i)15-s − 0.250·16-s + (−0.548 + 0.548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.845 + 0.534i$
Analytic conductor: \(13.5704\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :3/2),\ 0.845 + 0.534i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.89449 - 0.548428i\)
\(L(\frac12)\) \(\approx\) \(1.89449 - 0.548428i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 1.41i)T \)
5 \( 1 + (-11.1 + 0.923i)T \)
23 \( 1 + (-100. - 44.4i)T \)
good3 \( 1 + (-0.422 + 0.422i)T - 27iT^{2} \)
7 \( 1 + (-23.7 + 23.7i)T - 343iT^{2} \)
11 \( 1 - 53.9iT - 1.33e3T^{2} \)
13 \( 1 + (-33.6 + 33.6i)T - 2.19e3iT^{2} \)
17 \( 1 + (38.4 - 38.4i)T - 4.91e3iT^{2} \)
19 \( 1 + 54.7T + 6.85e3T^{2} \)
29 \( 1 - 42.1iT - 2.43e4T^{2} \)
31 \( 1 - 65.9T + 2.97e4T^{2} \)
37 \( 1 + (-146. + 146. i)T - 5.06e4iT^{2} \)
41 \( 1 - 424.T + 6.89e4T^{2} \)
43 \( 1 + (220. + 220. i)T + 7.95e4iT^{2} \)
47 \( 1 + (443. + 443. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-140. - 140. i)T + 1.48e5iT^{2} \)
59 \( 1 + 265. iT - 2.05e5T^{2} \)
61 \( 1 - 437. iT - 2.26e5T^{2} \)
67 \( 1 + (131. - 131. i)T - 3.00e5iT^{2} \)
71 \( 1 - 1.08e3T + 3.57e5T^{2} \)
73 \( 1 + (408. - 408. i)T - 3.89e5iT^{2} \)
79 \( 1 + 465.T + 4.93e5T^{2} \)
83 \( 1 + (528. + 528. i)T + 5.71e5iT^{2} \)
89 \( 1 + 223.T + 7.04e5T^{2} \)
97 \( 1 + (418. - 418. i)T - 9.12e5iT^{2} \)
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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

−11.27079504124645792923098420090, −10.60502323636751774259836300621, −10.05476399743901064027951402817, −8.711539527534426612177674410806, −7.80494339152753863329255617136, −6.91281967698197414567407363891, −5.17248586162286379116834607837, −4.22708536608593350543830764791, −2.20283344666965227481152756601, −1.31582776267523378838521793918, 1.23821114169703507061319787625, 2.70110909378718729599231708179, 4.75818741622169240039175637653, 5.96487629934999317539600391356, 6.45943704281928359172598783235, 8.273164757442713045404710965513, 8.856300395456148727247790762938, 9.494430622036088556910572232862, 11.07156127530915599264836247486, 11.43856706326634616302766095741

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