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Properties

Label 2-60e2-15.2-c1-0-27
Degree $2$
Conductor $3600$
Sign $0.391 + 0.920i$
Analytic cond. $28.7461$
Root an. cond. $5.36154$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.23 + 1.23i)7-s − 1.74i·11-s + (−0.236 + 0.236i)13-s + (4.57 − 4.57i)17-s − 6.47i·19-s + (2.82 + 2.82i)23-s − 0.333·29-s − 10.4·31-s + (2.23 + 2.23i)37-s − 7.07i·41-s + (−6.47 + 6.47i)43-s + (4.57 − 4.57i)47-s − 3.94i·49-s − 7.40·59-s + 1.52·61-s + ⋯
L(s)  = 1  + (0.467 + 0.467i)7-s − 0.527i·11-s + (−0.0654 + 0.0654i)13-s + (1.10 − 1.10i)17-s − 1.48i·19-s + (0.589 + 0.589i)23-s − 0.0619·29-s − 1.88·31-s + (0.367 + 0.367i)37-s − 1.10i·41-s + (−0.986 + 0.986i)43-s + (0.667 − 0.667i)47-s − 0.563i·49-s − 0.964·59-s + 0.195·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.391 + 0.920i$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ 0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.753985602\)
\(L(\frac12)\) \(\approx\) \(1.753985602\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.23 - 1.23i)T + 7iT^{2} \)
11 \( 1 + 1.74iT - 11T^{2} \)
13 \( 1 + (0.236 - 0.236i)T - 13iT^{2} \)
17 \( 1 + (-4.57 + 4.57i)T - 17iT^{2} \)
19 \( 1 + 6.47iT - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 + 0.333T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + (-2.23 - 2.23i)T + 37iT^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + (6.47 - 6.47i)T - 43iT^{2} \)
47 \( 1 + (-4.57 + 4.57i)T - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 7.40T + 59T^{2} \)
61 \( 1 - 1.52T + 61T^{2} \)
67 \( 1 + (10.4 + 10.4i)T + 67iT^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + (-9.47 + 9.47i)T - 73iT^{2} \)
79 \( 1 + 5.52iT - 79T^{2} \)
83 \( 1 + (-7.40 - 7.40i)T + 83iT^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (1 + i)T + 97iT^{2} \)
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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

−8.514621475090942549052130674648, −7.53383263722288199733476245560, −7.14314888961461468907799504097, −6.09311581733412801909006404351, −5.24151836121357625033124099918, −4.88112197294474864742605042258, −3.57743890642030676453051368397, −2.89232828231042936884743529602, −1.82391304632364675996670924005, −0.55758399378620294500428875838, 1.20957218817675284484238376562, 2.01566654623283635452301249777, 3.36195312223952631245061626982, 3.98810707217635525705480267664, 4.89633015408486574637964171357, 5.71302137937667533789585344627, 6.38978562533680057080288655866, 7.52225654754377383668747612146, 7.73153775097210652841982980269, 8.625411322268858425964871257105

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