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Properties

Label 2-3648-57.35-c0-0-1
Degree $2$
Conductor $3648$
Sign $-0.934 + 0.356i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
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  + (−0.939 + 0.342i)3-s + (−0.939 − 1.62i)7-s + (0.766 − 0.642i)9-s + (0.326 + 0.118i)13-s + (−0.5 − 0.866i)19-s + (1.43 + 1.20i)21-s + (−0.939 − 0.342i)25-s + (−0.500 + 0.866i)27-s + (0.766 + 1.32i)31-s − 1.53·37-s − 0.347·39-s + (0.266 − 1.50i)43-s + (−1.26 + 2.19i)49-s + (0.766 + 0.642i)57-s + (−0.0603 − 0.342i)61-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (−0.939 − 1.62i)7-s + (0.766 − 0.642i)9-s + (0.326 + 0.118i)13-s + (−0.5 − 0.866i)19-s + (1.43 + 1.20i)21-s + (−0.939 − 0.342i)25-s + (−0.500 + 0.866i)27-s + (0.766 + 1.32i)31-s − 1.53·37-s − 0.347·39-s + (0.266 − 1.50i)43-s + (−1.26 + 2.19i)49-s + (0.766 + 0.642i)57-s + (−0.0603 − 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $-0.934 + 0.356i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ -0.934 + 0.356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3059784263\)
\(L(\frac12)\) \(\approx\) \(0.3059784263\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + 1.53T + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
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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

−8.463616901736317214707214538200, −7.23280662967499051325247629758, −6.94925325893476854273822329469, −6.26976397159145473256869576178, −5.39849206740089576769385255950, −4.41601592470327829250695488857, −3.94489365934071503941948354272, −3.06850045705088212168166095894, −1.41623783353058845142069794457, −0.20352809628997342586575141276, 1.64669341687856409594742626332, 2.55527998819964757615233410287, 3.59532867358119026916143889671, 4.64579116809458894037019570361, 5.62675679308549822534162031389, 6.01452953539311896287431534670, 6.50514718613452964939576401932, 7.55192832927885377504029724026, 8.273860065037362320375755017828, 9.081832678476913103461938669379

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