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Properties

Label 2-1512-1512.349-c0-0-1
Degree $2$
Conductor $1512$
Sign $-0.835 - 0.549i$
Analytic cond. $0.754586$
Root an. cond. $0.868669$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.866 + 0.5i)3-s + (−0.939 − 0.342i)4-s + (−0.984 + 0.826i)5-s + (−0.642 + 0.766i)6-s + (0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.499 + 0.866i)9-s + (−0.642 − 1.11i)10-s + (−0.642 − 0.766i)12-s + (0.300 + 1.70i)13-s + (0.173 + 0.984i)14-s + (−1.26 + 0.223i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)18-s + (0.342 − 0.592i)19-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.866 + 0.5i)3-s + (−0.939 − 0.342i)4-s + (−0.984 + 0.826i)5-s + (−0.642 + 0.766i)6-s + (0.939 − 0.342i)7-s + (0.5 − 0.866i)8-s + (0.499 + 0.866i)9-s + (−0.642 − 1.11i)10-s + (−0.642 − 0.766i)12-s + (0.300 + 1.70i)13-s + (0.173 + 0.984i)14-s + (−1.26 + 0.223i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)18-s + (0.342 − 0.592i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(0.754586\)
Root analytic conductor: \(0.868669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :0),\ -0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120385078\)
\(L(\frac12)\) \(\approx\) \(1.120385078\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
good5 \( 1 + (0.984 - 0.826i)T + (0.173 - 0.984i)T^{2} \)
11 \( 1 + (-0.173 - 0.984i)T^{2} \)
13 \( 1 + (-0.300 - 1.70i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.342 + 0.592i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + (-0.766 - 0.642i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (1.85 - 0.673i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.300 + 1.70i)T + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.173 - 0.984i)T^{2} \)
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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

−9.797966151333718884327183454487, −8.905227936928137838912167777113, −8.354444345113343067639585999499, −7.49966493229543442365620090773, −7.16856118626904679481175143220, −6.12042610299801452667507156886, −4.66858404981101043099200852949, −4.29007961868071193869301706719, −3.43852569292816069791351550885, −1.89305935930304007011516054393, 0.955939526610726520653985970023, 2.03494176894730960570660012380, 3.24812393048910195722008732524, 3.94828876800952498529165780704, 4.89627674824550037028844919161, 5.84027968274509507466772002760, 7.66064234805396664880410893716, 7.980527338382261170815197188842, 8.381803023107319908205720495858, 9.233406604496555533308437553358

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