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Properties

Label 2-230-115.103-c1-0-2
Degree $2$
Conductor $230$
Sign $-0.516 - 0.856i$
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  + (−0.212 + 0.977i)2-s + (−0.409 + 0.306i)3-s + (−0.909 − 0.415i)4-s + (−0.0488 − 2.23i)5-s + (−0.212 − 0.464i)6-s + (−0.318 + 4.45i)7-s + (0.599 − 0.800i)8-s + (−0.771 + 2.62i)9-s + (2.19 + 0.427i)10-s + (−3.31 + 5.16i)11-s + (0.499 − 0.108i)12-s + (5.72 − 0.409i)13-s + (−4.28 − 1.25i)14-s + (0.704 + 0.899i)15-s + (0.654 + 0.755i)16-s + (0.199 − 0.533i)17-s + ⋯
L(s)  = 1  + (−0.150 + 0.690i)2-s + (−0.236 + 0.176i)3-s + (−0.454 − 0.207i)4-s + (−0.0218 − 0.999i)5-s + (−0.0866 − 0.189i)6-s + (−0.120 + 1.68i)7-s + (0.211 − 0.283i)8-s + (−0.257 + 0.875i)9-s + (0.694 + 0.135i)10-s + (−0.999 + 1.55i)11-s + (0.144 − 0.0313i)12-s + (1.58 − 0.113i)13-s + (−1.14 − 0.336i)14-s + (0.181 + 0.232i)15-s + (0.163 + 0.188i)16-s + (0.0482 − 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.516 - 0.856i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.516 - 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.429291 + 0.760720i\)
\(L(\frac12)\) \(\approx\) \(0.429291 + 0.760720i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.212 - 0.977i)T \)
5 \( 1 + (0.0488 + 2.23i)T \)
23 \( 1 + (-0.973 + 4.69i)T \)
good3 \( 1 + (0.409 - 0.306i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (0.318 - 4.45i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (3.31 - 5.16i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-5.72 + 0.409i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (-0.199 + 0.533i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (1.30 - 2.86i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-1.09 + 0.498i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.682 + 4.74i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-1.77 - 3.24i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (-3.17 + 0.933i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-3.83 - 5.11i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (-2.83 + 2.83i)T - 47iT^{2} \)
53 \( 1 + (3.02 + 0.216i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (0.100 + 0.0872i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-8.41 + 1.21i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (-3.29 - 0.717i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (4.82 - 3.09i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (0.0305 - 0.0113i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (2.54 - 2.93i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-15.8 + 8.63i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (-0.587 + 4.08i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-8.59 - 4.69i)T + (52.4 + 81.6i)T^{2} \)
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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

−12.73857798579118055035183769795, −11.70981109780041999019296255761, −10.43832360864884740896976939910, −9.387895155043430259229466520119, −8.474761287227727245473278645254, −7.87996689640166526711949003248, −6.13066913163962327284194944066, −5.38666060084755921885381272804, −4.50632128086427566495221310036, −2.20231075385489414847214858659, 0.810527960986368746907402475954, 3.20576081891914882889171784265, 3.82083311801919878829941467014, 5.82466884391956905524670546131, 6.81888326943579759836349205248, 7.914952566079435536926356671149, 9.060051947699250217838494478099, 10.43147521282735174852566158676, 10.92860245352632396302920286354, 11.40774509788830482577817633562

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