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Properties

Label 2-57e2-19.2-c0-0-2
Degree $2$
Conductor $3249$
Sign $-0.898 - 0.438i$
Analytic cond. $1.62146$
Root an. cond. $1.27336$
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)4-s + (−0.5 − 0.866i)7-s + (−1.11 − 1.32i)13-s + (0.766 + 0.642i)16-s + (−0.766 + 0.642i)25-s + (0.173 + 0.984i)28-s + (−1.5 + 0.866i)31-s + 1.73i·37-s + (0.939 − 0.342i)43-s + (0.592 + 1.62i)52-s + (−0.939 − 0.342i)61-s + (−0.500 − 0.866i)64-s + (−1.70 + 0.300i)67-s + (−0.766 − 0.642i)73-s + (−1.11 + 1.32i)79-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)4-s + (−0.5 − 0.866i)7-s + (−1.11 − 1.32i)13-s + (0.766 + 0.642i)16-s + (−0.766 + 0.642i)25-s + (0.173 + 0.984i)28-s + (−1.5 + 0.866i)31-s + 1.73i·37-s + (0.939 − 0.342i)43-s + (0.592 + 1.62i)52-s + (−0.939 − 0.342i)61-s + (−0.500 − 0.866i)64-s + (−1.70 + 0.300i)67-s + (−0.766 − 0.642i)73-s + (−1.11 + 1.32i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $-0.898 - 0.438i$
Analytic conductor: \(1.62146\)
Root analytic conductor: \(1.27336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3249} (838, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3249,\ (\ :0),\ -0.898 - 0.438i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09770022447\)
\(L(\frac12)\) \(\approx\) \(0.09770022447\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T^{2} \)
5 \( 1 + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.939 + 0.342i)T^{2} \)
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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

−8.427347326762540891954270788601, −7.61127251118026773830535322132, −7.10004806888946661442806147498, −5.96462625486963738301957819968, −5.31835686797723588310237591557, −4.57807633412518773519600820224, −3.69309169324620037127690417651, −2.94168876772325767304765489042, −1.39642556706200744112854699045, −0.06073485952792463367624484629, 1.97390689845240030146390198685, 2.84142433190567190140732683492, 3.99379254884095713505511341198, 4.48732399041520581191882621414, 5.52715141007913661055380113739, 6.06704149980976174944161509979, 7.23735686551164365640555416843, 7.69058350474119287472906965630, 8.782402120053638224308469772433, 9.258745197298653746433688944462

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