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Properties

Label 2-3640-3640.1299-c0-0-1
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 + (−1.70 − 0.984i)3-s + (0.499 − 0.866i)4-s + (−0.984 − 0.173i)5-s − 1.96·6-s + (−0.984 + 0.173i)7-s − 0.999i·8-s + (1.43 + 2.49i)9-s + (−0.939 + 0.342i)10-s + (−1.70 + 0.984i)12-s + i·13-s + (−0.766 + 0.642i)14-s + (1.50 + 1.26i)15-s + (−0.5 − 0.866i)16-s + (0.592 + 0.342i)17-s + (2.49 + 1.43i)18-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−1.70 − 0.984i)3-s + (0.499 − 0.866i)4-s + (−0.984 − 0.173i)5-s − 1.96·6-s + (−0.984 + 0.173i)7-s − 0.999i·8-s + (1.43 + 2.49i)9-s + (−0.939 + 0.342i)10-s + (−1.70 + 0.984i)12-s + i·13-s + (−0.766 + 0.642i)14-s + (1.50 + 1.26i)15-s + (−0.5 − 0.866i)16-s + (0.592 + 0.342i)17-s + (2.49 + 1.43i)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

\(L(\frac{1}{2})\) \(\approx\) \(0.6837463703\)
\(L(\frac12)\) \(\approx\) \(0.6837463703\)
\(L(1)\) 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 \( 1 + (0.984 + 0.173i)T \)
7 \( 1 + (0.984 - 0.173i)T \)
13 \( 1 - iT \)
good3 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.592 - 0.342i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.300 - 0.173i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.28iT - T^{2} \)
47 \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + 0.684T + T^{2} \)
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

−8.566364027081219708210619291516, −7.28526586163703498605421377817, −7.08374758128241403081913361058, −6.29105154699299345999899872332, −5.69392023494046674464787310475, −4.91758834268235454436887361180, −4.20554235623729442165245716791, −3.25706443200768762081674435097, −1.95850156509944504735205208313, −0.943366637788168739886539965856, 0.48475343453120020744448227184, 2.97430512521866271857957294027, 3.76278032134425059989801410473, 4.17101063816728272380783065879, 5.13302978241557675751494978705, 5.73135469675148650810594536941, 6.26766358125065324971255333548, 7.18514664422464695781131122258, 7.54913437328380924182219068997, 8.829302500013254805398218421741

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