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Properties

Label 2-230-115.114-c4-0-44
Degree $2$
Conductor $230$
Sign $-0.977 - 0.211i$
Analytic cond. $23.7750$
Root an. cond. $4.87597$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·2-s − 15.3i·3-s − 8.00·4-s + (24.9 − 1.69i)5-s − 43.4·6-s + 85.1·7-s + 22.6i·8-s − 155.·9-s + (−4.79 − 70.5i)10-s + 59.9i·11-s + 122. i·12-s − 296. i·13-s − 240. i·14-s + (−26.0 − 383. i)15-s + 64.0·16-s − 408.·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.70i·3-s − 0.500·4-s + (0.997 − 0.0678i)5-s − 1.20·6-s + 1.73·7-s + 0.353i·8-s − 1.91·9-s + (−0.0479 − 0.705i)10-s + 0.495i·11-s + 0.854i·12-s − 1.75i·13-s − 1.22i·14-s + (−0.115 − 1.70i)15-s + 0.250·16-s − 1.41·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.977 - 0.211i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.977 - 0.211i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.287472919\)
\(L(\frac12)\) \(\approx\) \(2.287472919\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
5 \( 1 + (-24.9 + 1.69i)T \)
23 \( 1 + (-146. + 508. i)T \)
good3 \( 1 + 15.3iT - 81T^{2} \)
7 \( 1 - 85.1T + 2.40e3T^{2} \)
11 \( 1 - 59.9iT - 1.46e4T^{2} \)
13 \( 1 + 296. iT - 2.85e4T^{2} \)
17 \( 1 + 408.T + 8.35e4T^{2} \)
19 \( 1 - 18.6iT - 1.30e5T^{2} \)
29 \( 1 + 895.T + 7.07e5T^{2} \)
31 \( 1 - 578.T + 9.23e5T^{2} \)
37 \( 1 + 622.T + 1.87e6T^{2} \)
41 \( 1 + 1.79e3T + 2.82e6T^{2} \)
43 \( 1 - 572.T + 3.41e6T^{2} \)
47 \( 1 - 2.21e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.71e3T + 7.89e6T^{2} \)
59 \( 1 - 2.38e3T + 1.21e7T^{2} \)
61 \( 1 - 2.02e3iT - 1.38e7T^{2} \)
67 \( 1 - 946.T + 2.01e7T^{2} \)
71 \( 1 - 3.82e3T + 2.54e7T^{2} \)
73 \( 1 + 86.9iT - 2.83e7T^{2} \)
79 \( 1 + 7.86e3iT - 3.89e7T^{2} \)
83 \( 1 + 8.55e3T + 4.74e7T^{2} \)
89 \( 1 - 1.08e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.51e4T + 8.85e7T^{2} \)
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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

−11.15040785848997601343941054938, −10.38786365736271814227766436558, −8.779261975515124666449288470806, −8.130411987134633689476257442328, −7.10158047472736003273955074294, −5.80090536113399176079187622334, −4.84466273331308643142244728296, −2.52766420510327222577666841817, −1.81845713390614189675821069602, −0.78111615107503029658074067327, 1.95741911970444597495671305474, 3.98865645055609587059443598886, 4.84312088230774235449674746282, 5.54336687723764251315056158480, 6.90121015399030569292549572182, 8.586782417992094805193331695079, 9.003991564321125582491169120189, 9.960204689553434556991856637199, 11.08438747425475230388277335262, 11.50109641516717223259951337980

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