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Properties

Label 2-57e2-171.67-c0-0-1
Degree $2$
Conductor $3249$
Sign $0.776 + 0.630i$
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  + (0.642 + 0.766i)2-s + (−0.642 − 0.766i)3-s + (0.766 − 0.642i)5-s + (0.173 − 0.984i)6-s + 7-s + (0.866 − 0.499i)8-s + (−0.173 + 0.984i)9-s + (0.984 + 0.173i)10-s + (0.5 − 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)14-s + (−0.984 − 0.173i)15-s + (0.939 + 0.342i)16-s + (−0.866 + 0.500i)18-s + (−0.642 − 0.766i)21-s + (0.984 − 0.173i)22-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.642 − 0.766i)3-s + (0.766 − 0.642i)5-s + (0.173 − 0.984i)6-s + 7-s + (0.866 − 0.499i)8-s + (−0.173 + 0.984i)9-s + (0.984 + 0.173i)10-s + (0.5 − 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)14-s + (−0.984 − 0.173i)15-s + (0.939 + 0.342i)16-s + (−0.866 + 0.500i)18-s + (−0.642 − 0.766i)21-s + (0.984 − 0.173i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3249\)    =    \(3^{2} \cdot 19^{2}\)
Sign: $0.776 + 0.630i$
Analytic conductor: \(1.62146\)
Root analytic conductor: \(1.27336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3249} (1777, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3249,\ (\ :0),\ 0.776 + 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.976868985\)
\(L(\frac12)\) \(\approx\) \(1.976868985\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.435600656820652202935685004320, −7.83531513714842721794740981663, −7.11790872596884508231022845742, −6.29271175231519682124500594598, −5.64013421605171804396830491855, −5.20965850207606028506237884281, −4.65712669219002156218858512658, −3.28086175450440497017471336957, −1.69968242447638502770920694400, −1.22646178390200968141256250067, 1.75970151453805307822597708042, 2.33094083713942036630560914682, 3.52009903605935007566659985082, 4.43488025879247029141618963240, 4.69839876431917125472767295960, 5.66875757650220144508670130383, 6.56535952342299081619968610698, 7.21396120487063478747909778135, 8.260190637850152282745270959893, 9.138111742761920786244593687545

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