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Properties

Label 2-230-115.67-c1-0-4
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  + (0.212 + 0.977i)2-s + (2.68 + 2.01i)3-s + (−0.909 + 0.415i)4-s + (−1.91 + 1.15i)5-s + (−1.39 + 3.05i)6-s + (−0.286 − 4.01i)7-s + (−0.599 − 0.800i)8-s + (2.32 + 7.92i)9-s + (−1.53 − 1.62i)10-s + (0.608 + 0.946i)11-s + (−3.27 − 0.713i)12-s + (1.75 + 0.125i)13-s + (3.85 − 1.13i)14-s + (−7.46 − 0.742i)15-s + (0.654 − 0.755i)16-s + (0.281 + 0.755i)17-s + ⋯
L(s)  = 1  + (0.150 + 0.690i)2-s + (1.55 + 1.16i)3-s + (−0.454 + 0.207i)4-s + (−0.855 + 0.517i)5-s + (−0.569 + 1.24i)6-s + (−0.108 − 1.51i)7-s + (−0.211 − 0.283i)8-s + (0.776 + 2.64i)9-s + (−0.485 − 0.513i)10-s + (0.183 + 0.285i)11-s + (−0.946 − 0.205i)12-s + (0.485 + 0.0347i)13-s + (1.03 − 0.302i)14-s + (−1.92 − 0.191i)15-s + (0.163 − 0.188i)16-s + (0.0683 + 0.183i)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: $\chi_{230} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.343 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01223 + 1.44762i\)
\(L(\frac12)\) \(\approx\) \(1.01223 + 1.44762i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.91 - 1.15i)T \)
23 \( 1 + (-4.13 + 2.43i)T \)
good3 \( 1 + (-2.68 - 2.01i)T + (0.845 + 2.87i)T^{2} \)
7 \( 1 + (0.286 + 4.01i)T + (-6.92 + 0.996i)T^{2} \)
11 \( 1 + (-0.608 - 0.946i)T + (-4.56 + 10.0i)T^{2} \)
13 \( 1 + (-1.75 - 0.125i)T + (12.8 + 1.85i)T^{2} \)
17 \( 1 + (-0.281 - 0.755i)T + (-12.8 + 11.1i)T^{2} \)
19 \( 1 + (2.21 + 4.85i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (2.62 + 1.19i)T + (18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.498 + 3.46i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (1.15 - 2.11i)T + (-20.0 - 31.1i)T^{2} \)
41 \( 1 + (-2.27 - 0.667i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (2.49 - 3.32i)T + (-12.1 - 41.2i)T^{2} \)
47 \( 1 + (-2.95 - 2.95i)T + 47iT^{2} \)
53 \( 1 + (6.73 - 0.481i)T + (52.4 - 7.54i)T^{2} \)
59 \( 1 + (-2.62 + 2.27i)T + (8.39 - 58.3i)T^{2} \)
61 \( 1 + (-11.6 - 1.67i)T + (58.5 + 17.1i)T^{2} \)
67 \( 1 + (1.29 - 0.282i)T + (60.9 - 27.8i)T^{2} \)
71 \( 1 + (12.9 + 8.34i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (6.76 + 2.52i)T + (55.1 + 47.8i)T^{2} \)
79 \( 1 + (6.05 + 6.98i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-3.94 - 2.15i)T + (44.8 + 69.8i)T^{2} \)
89 \( 1 + (0.442 + 3.07i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (16.3 - 8.91i)T + (52.4 - 81.6i)T^{2} \)
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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

−13.09150515872792022044613238193, −11.10683448764081577142964123556, −10.45274119390691100358641543458, −9.442565049044605773068777104739, −8.455663466616126959193732719380, −7.63286893403229369675308969133, −6.86068124355821409822035403734, −4.57923768870602046526170720553, −4.05346794474412131173543740485, −3.04428243140719078536068957490, 1.55721916285357693446568403082, 2.88334894813737550500240196224, 3.81997217323124138963180711079, 5.70528538229979181317540171602, 7.14753570830982016247547110529, 8.470034309953444829171551971786, 8.613351736266816198894460413138, 9.579801760405756454305383478002, 11.38927181441270716460913943587, 12.29566436349161701761491193429

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