Properties

Label 2-465-465.119-c1-0-16
Degree $2$
Conductor $465$
Sign $0.471 + 0.882i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
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  − 1.11·2-s + (−0.895 − 1.48i)3-s − 0.758·4-s + (−2.13 + 0.676i)5-s + (0.997 + 1.65i)6-s + (1.74 + 1.00i)7-s + 3.07·8-s + (−1.39 + 2.65i)9-s + (2.37 − 0.753i)10-s + (0.720 + 1.24i)11-s + (0.679 + 1.12i)12-s + (−0.994 − 1.72i)13-s + (−1.94 − 1.12i)14-s + (2.91 + 2.55i)15-s − 1.90·16-s + (0.447 + 0.258i)17-s + ⋯
L(s)  = 1  − 0.787·2-s + (−0.516 − 0.856i)3-s − 0.379·4-s + (−0.953 + 0.302i)5-s + (0.407 + 0.674i)6-s + (0.661 + 0.381i)7-s + 1.08·8-s + (−0.465 + 0.885i)9-s + (0.750 − 0.238i)10-s + (0.217 + 0.376i)11-s + (0.196 + 0.324i)12-s + (−0.275 − 0.477i)13-s + (−0.520 − 0.300i)14-s + (0.751 + 0.659i)15-s − 0.476·16-s + (0.108 + 0.0626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.471 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.471 + 0.882i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.471 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.453075 - 0.271614i\)
\(L(\frac12)\) \(\approx\) \(0.453075 - 0.271614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.895 + 1.48i)T \)
5 \( 1 + (2.13 - 0.676i)T \)
31 \( 1 + (2.21 + 5.10i)T \)
good2 \( 1 + 1.11T + 2T^{2} \)
7 \( 1 + (-1.74 - 1.00i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.720 - 1.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.994 + 1.72i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.447 - 0.258i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.42 - 2.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.25iT - 23T^{2} \)
29 \( 1 - 6.96T + 29T^{2} \)
37 \( 1 + (-3.15 + 5.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.50 + 3.17i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.20 - 2.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.36T + 47T^{2} \)
53 \( 1 + (-7.68 + 4.43i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.0 - 5.81i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 9.17iT - 61T^{2} \)
67 \( 1 + (-2.91 + 1.68i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.19 + 3.57i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.44 + 4.22i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.68 - 2.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.6 + 6.15i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.64T + 89T^{2} \)
97 \( 1 - 9.33iT - 97T^{2} \)
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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

−10.83074497980826682229109700103, −10.16951648030882908220902332130, −8.754986030445146499515536170786, −8.111134129399625759807301245789, −7.52487846120731610684917720484, −6.49247823351234255568696918363, −5.14856123767966126909935868899, −4.19253524160814704439986566068, −2.28725342248614546170529929142, −0.64447282635634015927591977333, 0.987246808257634674119054317427, 3.55951734210010394767350799948, 4.53112144972699987356933386007, 5.12618776477400032218330070702, 6.76687657669788118912029630983, 7.85256420959234536754466363076, 8.607625851803791866703472393143, 9.380138516790270619071044439012, 10.23119623430197335059984127675, 11.21166597456614094598952087934

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