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Properties

Label 2-230-115.103-c1-0-3
Degree $2$
Conductor $230$
Sign $0.805 - 0.592i$
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 + (−0.985 + 0.737i)3-s + (−0.909 − 0.415i)4-s + (1.29 + 1.82i)5-s + (0.511 + 1.11i)6-s + (−0.0864 + 1.20i)7-s + (−0.599 + 0.800i)8-s + (−0.418 + 1.42i)9-s + (2.05 − 0.877i)10-s + (−1.71 + 2.66i)11-s + (1.20 − 0.261i)12-s + (5.12 − 0.366i)13-s + (1.16 + 0.341i)14-s + (−2.62 − 0.842i)15-s + (0.654 + 0.755i)16-s + (1.27 − 3.41i)17-s + ⋯
L(s)  = 1  + (0.150 − 0.690i)2-s + (−0.569 + 0.426i)3-s + (−0.454 − 0.207i)4-s + (0.578 + 0.815i)5-s + (0.208 + 0.457i)6-s + (−0.0326 + 0.456i)7-s + (−0.211 + 0.283i)8-s + (−0.139 + 0.474i)9-s + (0.650 − 0.277i)10-s + (−0.516 + 0.804i)11-s + (0.347 − 0.0755i)12-s + (1.42 − 0.101i)13-s + (0.310 + 0.0912i)14-s + (−0.676 − 0.217i)15-s + (0.163 + 0.188i)16-s + (0.308 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.805 - 0.592i$
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: \((2,\ 230,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04927 + 0.344493i\)
\(L(\frac12)\) \(\approx\) \(1.04927 + 0.344493i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 - 1.82i)T \)
23 \( 1 + (-3.64 - 3.12i)T \)
good3 \( 1 + (0.985 - 0.737i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (0.0864 - 1.20i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (1.71 - 2.66i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-5.12 + 0.366i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (-1.27 + 3.41i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (0.552 - 1.21i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-0.381 + 0.174i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.00687 + 0.0478i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (3.33 + 6.11i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (7.66 - 2.25i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (7.09 + 9.47i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (0.789 - 0.789i)T - 47iT^{2} \)
53 \( 1 + (-10.3 - 0.739i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (1.43 + 1.24i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-3.22 + 0.463i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (-4.80 - 1.04i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (6.59 - 4.23i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-11.1 + 4.16i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (-9.78 + 11.2i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (5.45 - 2.97i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (-0.260 + 1.81i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (2.35 + 1.28i)T + (52.4 + 81.6i)T^{2} \)
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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

−12.06421868033397601650186090216, −11.19537380465746291547459006984, −10.51362040386325561034709575861, −9.798964216858283323662821702743, −8.630843703176616588672212187360, −7.17326311322464930220065841638, −5.80765413937170723136866914501, −5.08351962731608850647271603522, −3.47912479683046723461771213782, −2.11547978136924761399927917465, 1.02548930958865414580761736695, 3.60816706454865876848647200681, 5.10063340973948874132742055223, 6.04731692758011211683697487319, 6.71503233733909630044362084049, 8.294592603753859694218425426423, 8.808364034147732967353078298561, 10.18218143694910936372878099881, 11.22538815467858487641349924794, 12.34476162499924208266641501549

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