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Properties

Label 2-3120-12.11-c1-0-42
Degree $2$
Conductor $3120$
Sign $0.993 + 0.109i$
Analytic cond. $24.9133$
Root an. cond. $4.99132$
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  + (−1.58 + 0.695i)3-s − i·5-s − 4.55i·7-s + (2.03 − 2.20i)9-s + 2.12·11-s + 13-s + (0.695 + 1.58i)15-s + 7.54i·17-s + 2.37i·19-s + (3.16 + 7.22i)21-s + 0.524·23-s − 25-s + (−1.68 + 4.91i)27-s + 2.55i·29-s + 5.71i·31-s + ⋯
L(s)  = 1  + (−0.915 + 0.401i)3-s − 0.447i·5-s − 1.72i·7-s + (0.677 − 0.735i)9-s + 0.639·11-s + 0.277·13-s + (0.179 + 0.409i)15-s + 1.83i·17-s + 0.544i·19-s + (0.691 + 1.57i)21-s + 0.109·23-s − 0.200·25-s + (−0.324 + 0.945i)27-s + 0.474i·29-s + 1.02i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.993 + 0.109i$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ 0.993 + 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.395517409\)
\(L(\frac12)\) \(\approx\) \(1.395517409\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 0.695i)T \)
5 \( 1 + iT \)
13 \( 1 - T \)
good7 \( 1 + 4.55iT - 7T^{2} \)
11 \( 1 - 2.12T + 11T^{2} \)
17 \( 1 - 7.54iT - 17T^{2} \)
19 \( 1 - 2.37iT - 19T^{2} \)
23 \( 1 - 0.524T + 23T^{2} \)
29 \( 1 - 2.55iT - 29T^{2} \)
31 \( 1 - 5.71iT - 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 - 7.22iT - 41T^{2} \)
43 \( 1 + 9.46iT - 43T^{2} \)
47 \( 1 - 1.90T + 47T^{2} \)
53 \( 1 - 8.57iT - 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 15.4iT - 67T^{2} \)
71 \( 1 + 6.08T + 71T^{2} \)
73 \( 1 - 5.16T + 73T^{2} \)
79 \( 1 - 8.31iT - 79T^{2} \)
83 \( 1 + 3.50T + 83T^{2} \)
89 \( 1 + 9.61iT - 89T^{2} \)
97 \( 1 - 0.532T + 97T^{2} \)
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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

−8.669487336801351634620135634084, −7.898247042149047809895221490078, −7.01311725906691271370042795728, −6.41362019954448382082009018523, −5.68518817006178728194265202442, −4.66260499697121818746126873585, −4.00565482065484321772157043113, −3.58643462602493929151039364945, −1.54218898589148018609334258024, −0.850567204014183640246795840108, 0.72719534237957683095413518083, 2.21392443159151421294239111921, 2.74917376661729945875451357722, 4.15622887165332131325745014214, 5.12509309805694739796770200862, 5.69432104497807570078321364193, 6.38906004530041182799452059355, 7.03560159509932970390827435561, 7.85433727793708603486919646780, 8.764245348741918172221600695452

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