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Properties

Label 2-1512-63.59-c1-0-16
Degree $2$
Conductor $1512$
Sign $0.154 + 0.987i$
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  + 1.07·5-s + (−1.26 − 2.32i)7-s + 4.09i·11-s + (−3.69 − 2.13i)13-s + (−0.717 + 1.24i)17-s + (6.41 − 3.70i)19-s − 6.27i·23-s − 3.84·25-s + (8.09 − 4.67i)29-s + (5.96 − 3.44i)31-s + (−1.36 − 2.49i)35-s + (−0.453 − 0.785i)37-s + (−3.88 + 6.73i)41-s + (−6.32 − 10.9i)43-s + (4.21 − 7.30i)47-s + ⋯
L(s)  = 1  + 0.480·5-s + (−0.478 − 0.877i)7-s + 1.23i·11-s + (−1.02 − 0.592i)13-s + (−0.174 + 0.301i)17-s + (1.47 − 0.850i)19-s − 1.30i·23-s − 0.768·25-s + (1.50 − 0.868i)29-s + (1.07 − 0.618i)31-s + (−0.230 − 0.421i)35-s + (−0.0745 − 0.129i)37-s + (−0.607 + 1.05i)41-s + (−0.964 − 1.66i)43-s + (0.615 − 1.06i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.154 + 0.987i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.154 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.418719414\)
\(L(\frac12)\) \(\approx\) \(1.418719414\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 + 2.32i)T \)
good5 \( 1 - 1.07T + 5T^{2} \)
11 \( 1 - 4.09iT - 11T^{2} \)
13 \( 1 + (3.69 + 2.13i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.717 - 1.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.41 + 3.70i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.27iT - 23T^{2} \)
29 \( 1 + (-8.09 + 4.67i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.96 + 3.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.453 + 0.785i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.88 - 6.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.32 + 10.9i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.21 + 7.30i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.50 + 0.869i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.05 - 5.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.36 + 1.36i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.01 + 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.783iT - 71T^{2} \)
73 \( 1 + (1.95 + 1.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.817 - 1.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.48 + 7.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.71 + 2.97i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.05 + 2.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.639477961121280674793965544472, −8.464473036520071818155848982037, −7.51474328655087368015507886610, −6.96543031618962749428015121359, −6.14162708219610087439342260014, −4.94571897272316560022952562874, −4.40755027745453154220424485968, −3.08759180026251797643990117677, −2.15573108919275088837847091035, −0.57604210846505136524252004903, 1.38713596987743279435965663551, 2.74602685621426193818593139022, 3.38757652680892850650893091574, 4.85945282804478524341377613026, 5.59988504229820511498875330030, 6.26746193549291267536668636208, 7.19810108809250246855999967343, 8.159269332729660843354765581752, 8.941564325876503192347244478598, 9.702526858195804002914262962716

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