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Properties

Label 2-3640-3640.1133-c0-0-3
Degree $2$
Conductor $3640$
Sign $-0.0557 + 0.998i$
Analytic cond. $1.81659$
Root an. cond. $1.34781$
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  + (−0.866 + 0.5i)2-s + (0.448 − 1.67i)3-s + (0.499 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (0.448 + 1.67i)6-s + (−0.5 + 0.866i)7-s + 0.999i·8-s + (−1.73 − 1.00i)9-s + (0.707 − 0.707i)10-s + (−1.22 − 1.22i)12-s + (0.965 − 0.258i)13-s − 0.999i·14-s + 1.73i·15-s + (−0.5 − 0.866i)16-s + 2·18-s + (1.93 − 0.517i)19-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.448 − 1.67i)3-s + (0.499 − 0.866i)4-s + (−0.965 + 0.258i)5-s + (0.448 + 1.67i)6-s + (−0.5 + 0.866i)7-s + 0.999i·8-s + (−1.73 − 1.00i)9-s + (0.707 − 0.707i)10-s + (−1.22 − 1.22i)12-s + (0.965 − 0.258i)13-s − 0.999i·14-s + 1.73i·15-s + (−0.5 − 0.866i)16-s + 2·18-s + (1.93 − 0.517i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0557 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0557 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-0.0557 + 0.998i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ -0.0557 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7292930070\)
\(L(\frac12)\) \(\approx\) \(0.7292930070\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.965 + 0.258i)T \)
good3 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-1.93 + 0.517i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−8.298781950896819298656119584053, −7.79550481009320032187546528105, −7.25142478971156963447443475794, −6.51524342964901273704197257774, −6.02019852393484891795549181888, −5.10805145367215170634832995616, −3.38821983125637329494060620829, −2.79410279623681672522208789178, −1.71993964841485428406851071944, −0.63745537289132617597144991873, 1.14159291246487166727534091881, 2.89944693031466756849473602122, 3.59601269116903428465575682815, 3.89647407190100438220249355870, 4.73246536523287803188967647367, 5.85746571544672001450965823289, 7.04918306219546879117609830400, 7.76611258955554119295281699472, 8.338951198940727051847842040975, 9.155725283898775744710650985921

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