Properties

Label 2-648-648.115-c0-0-0
Degree $2$
Conductor $648$
Sign $-0.135 + 0.990i$
Analytic cond. $0.323394$
Root an. cond. $0.568677$
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  + (0.396 − 0.918i)2-s + (0.893 − 0.448i)3-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)6-s + (−0.939 + 0.342i)8-s + (0.597 − 0.802i)9-s + (1.14 + 0.754i)11-s + (−0.939 − 0.342i)12-s + (−0.0581 + 0.998i)16-s + (−1.52 + 1.27i)17-s + (−0.500 − 0.866i)18-s + (−1.28 − 1.07i)19-s + (1.14 − 0.754i)22-s + (−0.686 + 0.727i)24-s + (−0.993 + 0.116i)25-s + ⋯
L(s)  = 1  + (0.396 − 0.918i)2-s + (0.893 − 0.448i)3-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)6-s + (−0.939 + 0.342i)8-s + (0.597 − 0.802i)9-s + (1.14 + 0.754i)11-s + (−0.939 − 0.342i)12-s + (−0.0581 + 0.998i)16-s + (−1.52 + 1.27i)17-s + (−0.500 − 0.866i)18-s + (−1.28 − 1.07i)19-s + (1.14 − 0.754i)22-s + (−0.686 + 0.727i)24-s + (−0.993 + 0.116i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.135 + 0.990i$
Analytic conductor: \(0.323394\)
Root analytic conductor: \(0.568677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :0),\ -0.135 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.358444698\)
\(L(\frac12)\) \(\approx\) \(1.358444698\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.396 + 0.918i)T \)
3 \( 1 + (-0.893 + 0.448i)T \)
good5 \( 1 + (0.993 - 0.116i)T^{2} \)
7 \( 1 + (-0.893 - 0.448i)T^{2} \)
11 \( 1 + (-1.14 - 0.754i)T + (0.396 + 0.918i)T^{2} \)
13 \( 1 + (-0.973 - 0.230i)T^{2} \)
17 \( 1 + (1.52 - 1.27i)T + (0.173 - 0.984i)T^{2} \)
19 \( 1 + (1.28 + 1.07i)T + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.893 + 0.448i)T^{2} \)
29 \( 1 + (0.286 - 0.957i)T^{2} \)
31 \( 1 + (0.835 + 0.549i)T^{2} \)
37 \( 1 + (0.939 - 0.342i)T^{2} \)
41 \( 1 + (-0.707 - 1.64i)T + (-0.686 + 0.727i)T^{2} \)
43 \( 1 + (-0.707 + 0.355i)T + (0.597 - 0.802i)T^{2} \)
47 \( 1 + (0.835 - 0.549i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.0971 + 0.0639i)T + (0.396 - 0.918i)T^{2} \)
61 \( 1 + (0.0581 + 0.998i)T^{2} \)
67 \( 1 + (-1.16 + 1.56i)T + (-0.286 - 0.957i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (1.12 - 0.408i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.686 + 0.727i)T^{2} \)
83 \( 1 + (-0.137 + 0.318i)T + (-0.686 - 0.727i)T^{2} \)
89 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.0333 + 0.572i)T + (-0.993 - 0.116i)T^{2} \)
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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

−10.63116849617570988261757878145, −9.531090666022575825633733501488, −9.017472445890334369896893697320, −8.206066929899026227477642147435, −6.81923203993889202170599312770, −6.17880278888877936853554286166, −4.39069880315795297957636059758, −4.00511425290022442036443309756, −2.52149860568947923448255700738, −1.67990703480025224177193494747, 2.43172516855255334981062552482, 3.84330283078782636166832966621, 4.29391630568579765366176441828, 5.62528201469657750610477159018, 6.61719362663302468043039272866, 7.46647380153620511062042786542, 8.553845213891898032951248627334, 8.927902064064333592374109322316, 9.789076582628220501039432000450, 10.97793103197784265593241307545

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