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Properties

Label 2-1512-63.25-c1-0-10
Degree $2$
Conductor $1512$
Sign $0.977 - 0.211i$
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  + (0.263 − 0.455i)5-s + (−0.333 − 2.62i)7-s + (2.30 + 3.99i)11-s + (0.244 + 0.423i)13-s + (−2.75 + 4.77i)17-s + (1.83 + 3.18i)19-s + (−0.0269 + 0.0467i)23-s + (2.36 + 4.09i)25-s + (3.28 − 5.68i)29-s + 6.07·31-s + (−1.28 − 0.538i)35-s + (0.223 + 0.387i)37-s + (−2.52 − 4.36i)41-s + (2.84 − 4.93i)43-s + 9.19·47-s + ⋯
L(s)  = 1  + (0.117 − 0.203i)5-s + (−0.125 − 0.992i)7-s + (0.695 + 1.20i)11-s + (0.0678 + 0.117i)13-s + (−0.668 + 1.15i)17-s + (0.421 + 0.730i)19-s + (−0.00562 + 0.00974i)23-s + (0.472 + 0.818i)25-s + (0.609 − 1.05i)29-s + 1.09·31-s + (−0.216 − 0.0910i)35-s + (0.0367 + 0.0637i)37-s + (−0.394 − 0.682i)41-s + (0.434 − 0.752i)43-s + 1.34·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.977 - 0.211i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.977 - 0.211i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.764299701\)
\(L(\frac12)\) \(\approx\) \(1.764299701\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.333 + 2.62i)T \)
good5 \( 1 + (-0.263 + 0.455i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.30 - 3.99i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.244 - 0.423i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.75 - 4.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.83 - 3.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0269 - 0.0467i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.28 + 5.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.07T + 31T^{2} \)
37 \( 1 + (-0.223 - 0.387i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.52 + 4.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.84 + 4.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.19T + 47T^{2} \)
53 \( 1 + (-4.37 + 7.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.63T + 59T^{2} \)
61 \( 1 + 0.465T + 61T^{2} \)
67 \( 1 - 5.19T + 67T^{2} \)
71 \( 1 + 1.76T + 71T^{2} \)
73 \( 1 + (5.23 - 9.07i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + (4.49 - 7.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.05 - 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.22 + 9.04i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.640312500162713448016780321954, −8.706759466983836604285302745609, −7.87454945366340880161033825204, −6.99276222756214945277166156526, −6.45186966948277428129795961611, −5.31461304641600132582578623850, −4.23073167276632820917276497525, −3.80477228179444256045404232508, −2.21928015874330177744971984075, −1.11333511036242431950127419297, 0.875552874753532891257586207581, 2.56265718852012449142143254161, 3.12420660634992863196137184323, 4.49118078547348045372347238087, 5.35378947674546562599628920778, 6.28099862962559304043659359972, 6.80501580078667085574663106505, 7.969467723531940470890039122985, 8.961576609790427082584600716577, 9.074133275984971056657989746763

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