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Properties

Label 2-3640-3640.1133-c0-0-5
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 + (0.258 − 0.965i)3-s + (0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + (−0.258 − 0.965i)6-s + (0.5 − 0.866i)7-s − 0.999i·8-s + (0.707 + 0.707i)10-s + (−0.707 − 0.707i)12-s + (0.258 + 0.965i)13-s − 0.999i·14-s + 15-s + (−0.5 − 0.866i)16-s + (0.965 + 0.258i)20-s + (−0.707 − 0.707i)21-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.258 − 0.965i)3-s + (0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + (−0.258 − 0.965i)6-s + (0.5 − 0.866i)7-s − 0.999i·8-s + (0.707 + 0.707i)10-s + (−0.707 − 0.707i)12-s + (0.258 + 0.965i)13-s − 0.999i·14-s + 15-s + (−0.5 − 0.866i)16-s + (0.965 + 0.258i)20-s + (−0.707 − 0.707i)21-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

\(L(\frac{1}{2})\) \(\approx\) \(2.665803178\)
\(L(\frac12)\) \(\approx\) \(2.665803178\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.258 - 0.965i)T \)
good3 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.310165682928961437407025639832, −7.46298659260532866531226215339, −6.94205799836591180232925521731, −6.46352989056810205196403973614, −5.60152205552210218738244802099, −4.47292407679747039228062639374, −3.89944856153412306453959017772, −2.84821334543753394914289434488, −2.00952457895048230366644133967, −1.32006507941271110951348204809, 1.68736603593201402333932938825, 2.82377184730149391364554262065, 3.68360311094092675430666264967, 4.51549445713433194911111147056, 5.05845519899812800286214111842, 5.68337950521932724089120378949, 6.35469291287886848471738984584, 7.59996412588344321055226660609, 8.201700721402211550161426076890, 8.830143496747393175338049616240

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