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Properties

Label 2-3640-3640.2547-c0-0-6
Degree $2$
Conductor $3640$
Sign $-0.525 - 0.850i$
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  + (0.707 + 0.707i)2-s + (1.36 + 1.36i)3-s + 1.00i·4-s + (−0.258 − 0.965i)5-s + 1.93i·6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + 2.73i·9-s + (0.500 − 0.866i)10-s + (−1.36 + 1.36i)12-s + (0.707 + 0.707i)13-s + (0.866 − 0.500i)14-s + (0.965 − 1.67i)15-s − 1.00·16-s + (0.366 − 0.366i)17-s + (−1.93 + 1.93i)18-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (1.36 + 1.36i)3-s + 1.00i·4-s + (−0.258 − 0.965i)5-s + 1.93i·6-s + (0.258 − 0.965i)7-s + (−0.707 + 0.707i)8-s + 2.73i·9-s + (0.500 − 0.866i)10-s + (−1.36 + 1.36i)12-s + (0.707 + 0.707i)13-s + (0.866 − 0.500i)14-s + (0.965 − 1.67i)15-s − 1.00·16-s + (0.366 − 0.366i)17-s + (−1.93 + 1.93i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (2547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.866549247\)
\(L(\frac12)\) \(\approx\) \(2.866549247\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.93T + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.886001692603821492387506849008, −8.350818594167999260908969610778, −7.56559363162571450851362393985, −7.04498014330853044138983370092, −5.56259728234816931833829087527, −5.00855931520446760632569197744, −4.13510471100996976625645022768, −3.94968978336891418776496028771, −3.11185845303004155476728023921, −1.84514307332113406854250217043, 1.26177462664210825911641948816, 2.18163218281995968796512399133, 2.87262611800245047717255040018, 3.33635217738041177949473328266, 4.26528188374467119085087241415, 5.83496298585069320054831870734, 6.11091612513175992964465364843, 6.99751088454045620938975329432, 7.80876135072068182585619568306, 8.345658915983452875850149108092

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