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Properties

Label 2-2600-2600.2131-c0-0-3
Degree $2$
Conductor $2600$
Sign $0.985 - 0.166i$
Analytic cond. $1.29756$
Root an. cond. $1.13910$
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  + (0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.866i)5-s + (0.809 + 0.587i)6-s + 1.33·7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)10-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (0.413 + 1.27i)14-s + (−0.913 − 0.406i)15-s + (0.309 − 0.951i)16-s + (0.169 + 0.122i)17-s + (0.913 + 0.406i)20-s + (1.08 − 0.786i)21-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.866i)5-s + (0.809 + 0.587i)6-s + 1.33·7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)10-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (0.413 + 1.27i)14-s + (−0.913 − 0.406i)15-s + (0.309 − 0.951i)16-s + (0.169 + 0.122i)17-s + (0.913 + 0.406i)20-s + (1.08 − 0.786i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.985 - 0.166i$
Analytic conductor: \(1.29756\)
Root analytic conductor: \(1.13910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (2131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2600,\ (\ :0),\ 0.985 - 0.166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.769783353\)
\(L(\frac12)\) \(\approx\) \(1.769783353\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 - 1.33T + T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.564 + 1.73i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.33T + T^{2} \)
47 \( 1 + (-0.169 + 0.122i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (1.47 - 1.07i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−8.794654005765510197265018713533, −7.965812402314156859802200305920, −7.84514078429791429197004184712, −7.17527651105145007364994076130, −5.78392878068079831393948097871, −5.29824002310342934062048647594, −4.42325859647228440268252745254, −3.66412881170457221842255374332, −2.43419702547195889666490182996, −1.13281841170891587346248486963, 1.51330011689573335228657235293, 2.56498409482361125958096431462, 3.30667326977277782127220017602, 4.20595576689375830416824951184, 4.58648474974934061051980905485, 5.76594941231272729031560532152, 6.75145299354972870915829144772, 7.84393207410874481443298337243, 8.425037727565245227167278164438, 9.142420846205221024572407983303

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