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Properties

Label 2-230-115.103-c1-0-6
Degree $2$
Conductor $230$
Sign $0.943 - 0.330i$
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 + (0.627 − 0.470i)3-s + (−0.909 − 0.415i)4-s + (2.20 − 0.379i)5-s + (0.325 + 0.713i)6-s + (0.119 − 1.66i)7-s + (0.599 − 0.800i)8-s + (−0.671 + 2.28i)9-s + (−0.0977 + 2.23i)10-s + (2.43 − 3.79i)11-s + (−0.766 + 0.166i)12-s + (1.69 − 0.120i)13-s + (1.60 + 0.471i)14-s + (1.20 − 1.27i)15-s + (0.654 + 0.755i)16-s + (−1.34 + 3.59i)17-s + ⋯
L(s)  = 1  + (−0.150 + 0.690i)2-s + (0.362 − 0.271i)3-s + (−0.454 − 0.207i)4-s + (0.985 − 0.169i)5-s + (0.133 + 0.291i)6-s + (0.0451 − 0.630i)7-s + (0.211 − 0.283i)8-s + (−0.223 + 0.762i)9-s + (−0.0309 + 0.706i)10-s + (0.735 − 1.14i)11-s + (−0.221 + 0.0481i)12-s + (0.469 − 0.0335i)13-s + (0.429 + 0.125i)14-s + (0.311 − 0.328i)15-s + (0.163 + 0.188i)16-s + (−0.325 + 0.872i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.943 - 0.330i$
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: \((2,\ 230,\ (\ :1/2),\ 0.943 - 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39875 + 0.237498i\)
\(L(\frac12)\) \(\approx\) \(1.39875 + 0.237498i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 + 0.379i)T \)
23 \( 1 + (-2.54 + 4.06i)T \)
good3 \( 1 + (-0.627 + 0.470i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (-0.119 + 1.66i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (-2.43 + 3.79i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-1.69 + 0.120i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (1.34 - 3.59i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (1.30 - 2.85i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (8.84 - 4.04i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (-1.11 - 7.76i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (5.70 + 10.4i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (7.18 - 2.10i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (3.02 + 4.03i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (-2.25 + 2.25i)T - 47iT^{2} \)
53 \( 1 + (4.53 + 0.324i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (5.22 + 4.52i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-3.12 + 0.449i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (-0.601 - 0.130i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (9.57 - 6.15i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (3.89 - 1.45i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (-1.72 + 1.98i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (3.89 - 2.12i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (1.31 - 9.17i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (1.16 + 0.636i)T + (52.4 + 81.6i)T^{2} \)
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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

−12.66776430059294956892555269968, −10.92425556870351469988324008208, −10.39904675480258548454067867701, −8.856460125431356392389941562490, −8.562410465658468224926915805509, −7.17862419011382331889954318195, −6.19848030870499709641773871419, −5.23398384627540518573771211280, −3.63199250839600962857056466638, −1.63404377434069657853003131615, 1.86843579737884240791286186715, 3.13596092402165611883680969204, 4.57986098042318759999549562993, 5.93251309489284165171515913558, 7.09361453519242869058327797497, 8.761679878579436313049018747521, 9.400611893433042384372330056004, 9.910663297800867933681718604228, 11.32568348840209214316211934600, 11.96905569759884450210927877338

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