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Properties

Label 2-3640-3640.1077-c0-0-5
Degree $2$
Conductor $3640$
Sign $0.997 - 0.0685i$
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 + (1.67 + 0.448i)3-s + (0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + (1.67 − 0.448i)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 + (−0.258 − 0.965i)13-s + 0.999i·14-s + 1.73i·15-s + (−0.5 − 0.866i)16-s + 2·18-s + (−0.517 − 1.93i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (1.67 + 0.448i)3-s + (0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + (1.67 − 0.448i)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 + (−0.258 − 0.965i)13-s + 0.999i·14-s + 1.73i·15-s + (−0.5 − 0.866i)16-s + 2·18-s + (−0.517 − 1.93i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.997 - 0.0685i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ 0.997 - 0.0685i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.626724954\)
\(L(\frac12)\) \(\approx\) \(3.626724954\)
\(L(1)\) 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 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.517 + 1.93i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 - 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
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

−9.029679313925975684660486885925, −7.950898565355533935789825762051, −7.26529343846738319045185378829, −6.47686553133372029496354147529, −5.58450551504646396435288025959, −4.77798181599008564588720509168, −3.66759604291755146863957287343, −3.15563166225950592185194139766, −2.59181573044582365003316047898, −1.94538507105415355683985911479, 1.59694448533330585774424679501, 2.33374687638527455712445886458, 3.38109539021505250913335390353, 4.18864052551230891661379604831, 4.46526540224803789287014268938, 5.88912731083879501888525651392, 6.57123417224604019890991692475, 7.27614358579463618815680552097, 8.075435110131460009622736360823, 8.396701365029869471682274205453

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