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Properties

Label 2-3640-3640.1299-c0-0-11
Degree $2$
Conductor $3640$
Sign $0.832 + 0.553i$
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 + (0.592 + 0.342i)3-s + (0.499 − 0.866i)4-s + (0.342 + 0.939i)5-s + 0.684·6-s + (0.342 − 0.939i)7-s − 0.999i·8-s + (−0.266 − 0.460i)9-s + (0.766 + 0.642i)10-s + (0.592 − 0.342i)12-s + i·13-s + (−0.173 − 0.984i)14-s + (−0.118 + 0.673i)15-s + (−0.5 − 0.866i)16-s + (1.11 + 0.642i)17-s + (−0.460 − 0.266i)18-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.592 + 0.342i)3-s + (0.499 − 0.866i)4-s + (0.342 + 0.939i)5-s + 0.684·6-s + (0.342 − 0.939i)7-s − 0.999i·8-s + (−0.266 − 0.460i)9-s + (0.766 + 0.642i)10-s + (0.592 − 0.342i)12-s + i·13-s + (−0.173 − 0.984i)14-s + (−0.118 + 0.673i)15-s + (−0.5 − 0.866i)16-s + (1.11 + 0.642i)17-s + (−0.460 − 0.266i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.843382547\)
\(L(\frac12)\) \(\approx\) \(2.843382547\)
\(L(1)\) 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 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.342 + 0.939i)T \)
13 \( 1 - iT \)
good3 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.96iT - T^{2} \)
47 \( 1 + (1.32 - 0.766i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.28T + T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.845321343946439834960178755326, −7.70858700891086026477854345926, −7.05796028532957916549938828239, −6.34961266767573401117043602204, −5.64146015946302456297820430436, −4.60173485690047803681043803359, −3.71544457834365680455414188238, −3.42231606968631711124890462004, −2.36268340367464506238198205335, −1.40201451021297382301367546296, 1.56784169897467978275311095558, 2.58627703874167244424362904163, 3.14058534105096579068495559997, 4.41944861070107980185257124262, 5.18173417695827830761328263806, 5.62773282345231225640261338787, 6.29624847943120148755206319098, 7.69273168004459186776465809183, 7.86639186519483758438380639540, 8.482210289260854108685679855807

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