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Properties

Label 2-1512-63.25-c1-0-22
Degree $2$
Conductor $1512$
Sign $-0.579 + 0.814i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (1.33 − 2.31i)5-s + (0.581 − 2.58i)7-s + (0.682 + 1.18i)11-s + (−2.75 − 4.77i)13-s + (1.23 − 2.14i)17-s + (−2.19 − 3.80i)19-s + (−2.34 + 4.06i)23-s + (−1.05 − 1.83i)25-s + (−2.94 + 5.10i)29-s + 3.11·31-s + (−5.18 − 4.78i)35-s + (−3.15 − 5.46i)37-s + (−1.38 − 2.40i)41-s + (−4.87 + 8.45i)43-s + 10.0·47-s + ⋯
L(s)  = 1  + (0.596 − 1.03i)5-s + (0.219 − 0.975i)7-s + (0.205 + 0.356i)11-s + (−0.764 − 1.32i)13-s + (0.300 − 0.520i)17-s + (−0.503 − 0.872i)19-s + (−0.488 + 0.846i)23-s + (−0.211 − 0.366i)25-s + (−0.547 + 0.948i)29-s + 0.559·31-s + (−0.877 − 0.809i)35-s + (−0.518 − 0.898i)37-s + (−0.216 − 0.375i)41-s + (−0.744 + 1.28i)43-s + 1.46·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.579 + 0.814i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.579 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.544280292\)
\(L(\frac12)\) \(\approx\) \(1.544280292\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-0.581 + 2.58i)T \)
good5 \( 1 + (-1.33 + 2.31i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.682 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.75 + 4.77i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.23 + 2.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.19 + 3.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.34 - 4.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.94 - 5.10i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.11T + 31T^{2} \)
37 \( 1 + (3.15 + 5.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.38 + 2.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.87 - 8.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + (-1.47 + 2.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.55T + 59T^{2} \)
61 \( 1 - 1.32T + 61T^{2} \)
67 \( 1 - 8.29T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (1.11 - 1.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + (5.15 - 8.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.73 + 13.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.55 - 4.42i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.295482256982131396832705584305, −8.419861007889563940112992601175, −7.55128236858673608675564062926, −6.95501425829027231439121726750, −5.64819720840385943901459606702, −5.07620474382661570726544614191, −4.27351009957635358584435098569, −3.07348790194659214926258897976, −1.72637249232789305772871141526, −0.58795722748677116323742177282, 1.90954212862365868532575110322, 2.50075554236267633325993964611, 3.72259993621824211896554310038, 4.77391535228306856398243714740, 5.96192876941456337686479313754, 6.31703271918285431460042743351, 7.22146915454677275100401932946, 8.277875931998781851982076463234, 8.929740870028063264063587158829, 9.896773317114161279806560352910

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