Properties

Label 2-230-115.29-c1-0-11
Degree $2$
Conductor $230$
Sign $0.343 + 0.939i$
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  + (0.909 + 0.415i)2-s + (0.768 − 2.61i)3-s + (0.654 + 0.755i)4-s + (−1.08 − 1.95i)5-s + (1.78 − 2.06i)6-s + (−1.27 − 0.182i)7-s + (0.281 + 0.959i)8-s + (−3.73 − 2.40i)9-s + (−0.169 − 2.22i)10-s + (1.03 + 2.27i)11-s + (2.48 − 1.13i)12-s + (1.02 − 0.146i)13-s + (−1.07 − 0.693i)14-s + (−5.95 + 1.32i)15-s + (−0.142 + 0.989i)16-s + (3.98 + 3.45i)17-s + ⋯
L(s)  = 1  + (0.643 + 0.293i)2-s + (0.443 − 1.51i)3-s + (0.327 + 0.377i)4-s + (−0.483 − 0.875i)5-s + (0.729 − 0.841i)6-s + (−0.480 − 0.0690i)7-s + (0.0996 + 0.339i)8-s + (−1.24 − 0.800i)9-s + (−0.0535 − 0.705i)10-s + (0.313 + 0.686i)11-s + (0.716 − 0.327i)12-s + (0.283 − 0.0406i)13-s + (−0.288 − 0.185i)14-s + (−1.53 + 0.341i)15-s + (−0.0355 + 0.247i)16-s + (0.965 + 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.343 + 0.939i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.343 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46552 - 1.02428i\)
\(L(\frac12)\) \(\approx\) \(1.46552 - 1.02428i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.909 - 0.415i)T \)
5 \( 1 + (1.08 + 1.95i)T \)
23 \( 1 + (-2.14 + 4.28i)T \)
good3 \( 1 + (-0.768 + 2.61i)T + (-2.52 - 1.62i)T^{2} \)
7 \( 1 + (1.27 + 0.182i)T + (6.71 + 1.97i)T^{2} \)
11 \( 1 + (-1.03 - 2.27i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-1.02 + 0.146i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (-3.98 - 3.45i)T + (2.41 + 16.8i)T^{2} \)
19 \( 1 + (-0.540 - 0.623i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-1.11 + 1.28i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-7.72 + 2.26i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (5.04 - 7.85i)T + (-15.3 - 33.6i)T^{2} \)
41 \( 1 + (3.34 - 2.15i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (2.74 - 9.34i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + (3.94 + 0.567i)T + (50.8 + 14.9i)T^{2} \)
59 \( 1 + (-1.11 - 7.73i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (3.40 - 0.998i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (13.0 + 5.97i)T + (43.8 + 50.6i)T^{2} \)
71 \( 1 + (-2.65 + 5.82i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (12.6 - 10.9i)T + (10.3 - 72.2i)T^{2} \)
79 \( 1 + (-1.01 - 7.07i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (2.41 - 3.75i)T + (-34.4 - 75.4i)T^{2} \)
89 \( 1 + (-6.62 - 1.94i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (8.57 + 13.3i)T + (-40.2 + 88.2i)T^{2} \)
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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

−12.24399037747675850922612965701, −11.78158353689101189713622589304, −10.01314901321881471291109629234, −8.540682637378214203252368991091, −7.986149733909733626220521931470, −6.93419764977677748728846672410, −6.10178301831272397435097805935, −4.60878743301709406520763815939, −3.16077276545896909579395910970, −1.43458001842136955744581681493, 3.10486179902995604341469529262, 3.48256336352627775445123962676, 4.77359925524147700576038972933, 5.98833422394784346047721968925, 7.32246752052280747951934659318, 8.761278056907962627474278839343, 9.745470107323103969976077409203, 10.50811698287011723746079287453, 11.29557748302670680564425120270, 12.15309675530727281431802197369

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