Properties

Label 2-230-115.107-c1-0-7
Degree $2$
Conductor $230$
Sign $0.916 + 0.399i$
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  + (0.800 − 0.599i)2-s + (1.22 + 0.456i)3-s + (0.281 − 0.959i)4-s + (2.22 + 0.215i)5-s + (1.25 − 0.367i)6-s + (−2.43 + 0.528i)7-s + (−0.349 − 0.936i)8-s + (−0.979 − 0.849i)9-s + (1.91 − 1.16i)10-s + (3.01 − 0.433i)11-s + (0.782 − 1.04i)12-s + (−0.464 + 2.13i)13-s + (−1.62 + 1.88i)14-s + (2.62 + 1.27i)15-s + (−0.841 − 0.540i)16-s + (−2.42 + 4.44i)17-s + ⋯
L(s)  = 1  + (0.566 − 0.423i)2-s + (0.706 + 0.263i)3-s + (0.140 − 0.479i)4-s + (0.995 + 0.0963i)5-s + (0.511 − 0.150i)6-s + (−0.918 + 0.199i)7-s + (−0.123 − 0.331i)8-s + (−0.326 − 0.283i)9-s + (0.604 − 0.367i)10-s + (0.909 − 0.130i)11-s + (0.225 − 0.301i)12-s + (−0.128 + 0.591i)13-s + (−0.435 + 0.502i)14-s + (0.677 + 0.330i)15-s + (−0.210 − 0.135i)16-s + (−0.589 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.916 + 0.399i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.916 + 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01443 - 0.420195i\)
\(L(\frac12)\) \(\approx\) \(2.01443 - 0.420195i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.800 + 0.599i)T \)
5 \( 1 + (-2.22 - 0.215i)T \)
23 \( 1 + (-1.66 + 4.49i)T \)
good3 \( 1 + (-1.22 - 0.456i)T + (2.26 + 1.96i)T^{2} \)
7 \( 1 + (2.43 - 0.528i)T + (6.36 - 2.90i)T^{2} \)
11 \( 1 + (-3.01 + 0.433i)T + (10.5 - 3.09i)T^{2} \)
13 \( 1 + (0.464 - 2.13i)T + (-11.8 - 5.40i)T^{2} \)
17 \( 1 + (2.42 - 4.44i)T + (-9.19 - 14.3i)T^{2} \)
19 \( 1 + (5.85 + 1.71i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-1.38 - 4.71i)T + (-24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.735 + 1.60i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (0.219 + 3.07i)T + (-36.6 + 5.26i)T^{2} \)
41 \( 1 + (2.35 + 2.71i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (3.49 - 9.36i)T + (-32.4 - 28.1i)T^{2} \)
47 \( 1 + (1.99 + 1.99i)T + 47iT^{2} \)
53 \( 1 + (-2.52 - 11.6i)T + (-48.2 + 22.0i)T^{2} \)
59 \( 1 + (3.76 + 5.85i)T + (-24.5 + 53.6i)T^{2} \)
61 \( 1 + (2.41 + 1.10i)T + (39.9 + 46.1i)T^{2} \)
67 \( 1 + (-5.36 - 7.16i)T + (-18.8 + 64.2i)T^{2} \)
71 \( 1 + (-1.05 + 7.33i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-8.78 + 4.79i)T + (39.4 - 61.4i)T^{2} \)
79 \( 1 + (-11.0 + 7.11i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (12.5 - 0.894i)T + (82.1 - 11.8i)T^{2} \)
89 \( 1 + (0.366 + 0.802i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-17.4 - 1.24i)T + (96.0 + 13.8i)T^{2} \)
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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

−12.44396690460065353418948047372, −11.12426433142525136329002720820, −10.18404560924588219765718416934, −9.183096424027674844392735605923, −8.741496171239493025461372277681, −6.51265239346160960156732981865, −6.23667898601286500915253811467, −4.47887004031702132788048865695, −3.28182319992224137911053300840, −2.11698134456436157398634997093, 2.26695507139995154032368517533, 3.47028134888411750391828042583, 5.04806278757041716218521211514, 6.24930548924133576800678391300, 7.00890996747828241835274936278, 8.346684684840904711374916203214, 9.236802782302390273268715691917, 10.14085016807594294829797781816, 11.48407399011240345164607205433, 12.70896708262588218071008561953

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