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Properties

Label 2-1512-7.2-c1-0-30
Degree $2$
Conductor $1512$
Sign $-0.984 + 0.174i$
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  + (−0.366 + 0.633i)5-s + (2.12 − 1.57i)7-s + (−1.92 − 3.33i)11-s − 4.87·13-s + (−3.09 − 5.36i)17-s + (−1.39 + 2.41i)19-s + (−3.69 + 6.40i)23-s + (2.23 + 3.86i)25-s − 9.78·29-s + (3.15 + 5.46i)31-s + (0.216 + 1.92i)35-s + (2.82 − 4.88i)37-s − 1.50·41-s − 12.1·43-s + (2.95 − 5.12i)47-s + ⋯
L(s)  = 1  + (−0.163 + 0.283i)5-s + (0.804 − 0.593i)7-s + (−0.580 − 1.00i)11-s − 1.35·13-s + (−0.751 − 1.30i)17-s + (−0.320 + 0.555i)19-s + (−0.771 + 1.33i)23-s + (0.446 + 0.773i)25-s − 1.81·29-s + (0.566 + 0.982i)31-s + (0.0365 + 0.325i)35-s + (0.463 − 0.803i)37-s − 0.234·41-s − 1.85·43-s + (0.431 − 0.747i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.984 + 0.174i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.984 + 0.174i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2851618580\)
\(L(\frac12)\) \(\approx\) \(0.2851618580\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 1.57i)T \)
good5 \( 1 + (0.366 - 0.633i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.92 + 3.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.87T + 13T^{2} \)
17 \( 1 + (3.09 + 5.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.39 - 2.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.69 - 6.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.78T + 29T^{2} \)
31 \( 1 + (-3.15 - 5.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.82 + 4.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.50T + 41T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 + (-2.95 + 5.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.43 - 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.19 + 5.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.29 + 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.834 + 1.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + (-0.720 - 1.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.55 - 6.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.98T + 83T^{2} \)
89 \( 1 + (3.62 - 6.27i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−9.121940069474751556875501823740, −8.155067349593826293470442700829, −7.45100223053700057215777123189, −6.95496566140145736896972964787, −5.56827649949375317349544752295, −5.04196981748832614530996192036, −3.94881070073955876291767163420, −2.96020917831671011774302728021, −1.78779798988716656667008847571, −0.10230181979852868072474992211, 1.95522782397663878498349072363, 2.52597626993924625247688386114, 4.32809490035667247325333227542, 4.65297231110785999919762993050, 5.66402526011308192930449968405, 6.65834016418618265334513500656, 7.57226985919516438228540575841, 8.284637435381455724564886664684, 8.891574739715139783621373908128, 9.989132045217627527882105692904

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