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Properties

Label 2-230-5.2-c2-0-17
Degree $2$
Conductor $230$
Sign $-0.581 + 0.813i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−0.454 + 0.454i)3-s + 2i·4-s + (4.61 − 1.92i)5-s + 0.908·6-s + (−8.21 − 8.21i)7-s + (2 − 2i)8-s + 8.58i·9-s + (−6.54 − 2.68i)10-s − 1.80·11-s + (−0.908 − 0.908i)12-s + (12.1 − 12.1i)13-s + 16.4i·14-s + (−1.21 + 2.97i)15-s − 4·16-s + (−13.2 − 13.2i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.151 + 0.151i)3-s + 0.5i·4-s + (0.922 − 0.385i)5-s + 0.151·6-s + (−1.17 − 1.17i)7-s + (0.250 − 0.250i)8-s + 0.954i·9-s + (−0.654 − 0.268i)10-s − 0.164·11-s + (−0.0756 − 0.0756i)12-s + (0.934 − 0.934i)13-s + 1.17i·14-s + (−0.0812 + 0.198i)15-s − 0.250·16-s + (−0.778 − 0.778i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.581 + 0.813i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.418836 - 0.814729i\)
\(L(\frac12)\) \(\approx\) \(0.418836 - 0.814729i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 + (-4.61 + 1.92i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good3 \( 1 + (0.454 - 0.454i)T - 9iT^{2} \)
7 \( 1 + (8.21 + 8.21i)T + 49iT^{2} \)
11 \( 1 + 1.80T + 121T^{2} \)
13 \( 1 + (-12.1 + 12.1i)T - 169iT^{2} \)
17 \( 1 + (13.2 + 13.2i)T + 289iT^{2} \)
19 \( 1 + 13.8iT - 361T^{2} \)
29 \( 1 + 47.4iT - 841T^{2} \)
31 \( 1 + 56.5T + 961T^{2} \)
37 \( 1 + (-3.64 - 3.64i)T + 1.36e3iT^{2} \)
41 \( 1 + 8.30T + 1.68e3T^{2} \)
43 \( 1 + (-17.7 + 17.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-8.93 - 8.93i)T + 2.20e3iT^{2} \)
53 \( 1 + (-24.9 + 24.9i)T - 2.80e3iT^{2} \)
59 \( 1 - 6.35iT - 3.48e3T^{2} \)
61 \( 1 - 98.9T + 3.72e3T^{2} \)
67 \( 1 + (16.0 + 16.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 110.T + 5.04e3T^{2} \)
73 \( 1 + (72.6 - 72.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 81.2iT - 6.24e3T^{2} \)
83 \( 1 + (-70.5 + 70.5i)T - 6.88e3iT^{2} \)
89 \( 1 - 131. iT - 7.92e3T^{2} \)
97 \( 1 + (-79.5 - 79.5i)T + 9.40e3iT^{2} \)
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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

−11.28538256755574521356211851403, −10.53435607686303489217690172786, −9.873734801787900087802210204781, −8.982733847649075454665832150021, −7.73067477359653731673885797616, −6.62401511041252020640466266807, −5.35614555701117146517687563253, −3.91844640474155854163158284621, −2.43889114193956663052080015577, −0.58066375882092270764014845382, 1.87324552038256509323137293877, 3.50926330561058036903968444025, 5.65650756310364488297411180748, 6.25840020115332203688849246089, 6.93647887505406179198588197663, 8.893008962848016132599845766384, 9.072936215000150180383995178352, 10.16231225524594323953455212184, 11.20782747166312589113680763291, 12.48622100501617399116292324727

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