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Properties

Label 2-1512-63.25-c1-0-0
Degree $2$
Conductor $1512$
Sign $-0.709 - 0.705i$
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  + (−0.918 + 1.59i)5-s + (−0.361 − 2.62i)7-s + (−1.54 − 2.68i)11-s + (2.40 + 4.16i)13-s + (−1.87 + 3.24i)17-s + (−2.71 − 4.70i)19-s + (−3.97 + 6.89i)23-s + (0.813 + 1.40i)25-s + (0.325 − 0.563i)29-s + 1.03·31-s + (4.50 + 1.83i)35-s + (0.873 + 1.51i)37-s + (−2.52 − 4.36i)41-s + (−6.09 + 10.5i)43-s + 4.61·47-s + ⋯
L(s)  = 1  + (−0.410 + 0.711i)5-s + (−0.136 − 0.990i)7-s + (−0.466 − 0.808i)11-s + (0.666 + 1.15i)13-s + (−0.453 + 0.786i)17-s + (−0.622 − 1.07i)19-s + (−0.829 + 1.43i)23-s + (0.162 + 0.281i)25-s + (0.0604 − 0.104i)29-s + 0.186·31-s + (0.760 + 0.309i)35-s + (0.143 + 0.248i)37-s + (−0.393 − 0.682i)41-s + (−0.929 + 1.61i)43-s + 0.672·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.709 - 0.705i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.709 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6123136950\)
\(L(\frac12)\) \(\approx\) \(0.6123136950\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.361 + 2.62i)T \)
good5 \( 1 + (0.918 - 1.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.54 + 2.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.87 - 3.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.71 + 4.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.97 - 6.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.325 + 0.563i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.03T + 31T^{2} \)
37 \( 1 + (-0.873 - 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.52 + 4.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.09 - 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.61T + 47T^{2} \)
53 \( 1 + (4.55 - 7.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.79T + 59T^{2} \)
61 \( 1 + 4.81T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 - 5.00T + 71T^{2} \)
73 \( 1 + (1.81 - 3.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 + (3.83 - 6.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.76 - 9.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.04 - 1.80i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.836984246029933207150585051536, −8.924782814413200175122733453721, −8.128218912206622742714048585531, −7.28579559672557891885959145038, −6.62411014058797412032210064569, −5.88701538634692788032419764732, −4.51434473295870697293694536632, −3.82164497379051396035567019265, −2.95438573127422286240971317541, −1.50152736652017719441902811481, 0.23734994505371332974361434345, 1.94744561100907289715647000971, 2.96536209258642349383044857445, 4.19708674610312290997571802440, 5.00010458303719611503768799998, 5.81492883805851386180600596126, 6.65424608869631129893589907273, 7.84278920998385155855116890952, 8.409589053741410420573621171354, 8.945099729721354150627616262592

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