Properties

Label 2-3120-13.12-c1-0-32
Degree $2$
Conductor $3120$
Sign $0.155 + 0.987i$
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  − 3-s + i·5-s + 0.561i·7-s + 9-s + 1.43i·11-s + (3.56 − 0.561i)13-s i·15-s − 4.56·17-s − 7.12i·19-s − 0.561i·21-s − 1.43·23-s − 25-s − 27-s − 8.24·29-s − 6i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447i·5-s + 0.212i·7-s + 0.333·9-s + 0.433i·11-s + (0.987 − 0.155i)13-s − 0.258i·15-s − 1.10·17-s − 1.63i·19-s − 0.122i·21-s − 0.299·23-s − 0.200·25-s − 0.192·27-s − 1.53·29-s − 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.155 + 0.987i$
Analytic conductor: \(24.9133\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :1/2),\ 0.155 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9173808025\)
\(L(\frac12)\) \(\approx\) \(0.9173808025\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - iT \)
13 \( 1 + (-3.56 + 0.561i)T \)
good7 \( 1 - 0.561iT - 7T^{2} \)
11 \( 1 - 1.43iT - 11T^{2} \)
17 \( 1 + 4.56T + 17T^{2} \)
19 \( 1 + 7.12iT - 19T^{2} \)
23 \( 1 + 1.43T + 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 + 6iT - 31T^{2} \)
37 \( 1 + 1.43iT - 37T^{2} \)
41 \( 1 - 4.56iT - 41T^{2} \)
43 \( 1 - 1.12T + 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 - 11.1iT - 67T^{2} \)
71 \( 1 + 3.68iT - 71T^{2} \)
73 \( 1 + 2.87iT - 73T^{2} \)
79 \( 1 - 1.43T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−8.717698043017851906382234292924, −7.52813581862598510070605825048, −7.04765271145178157031459906742, −6.18752001127751869210863094341, −5.63828381436253181557421972529, −4.58988885221402889629209918859, −3.94367320519161718088337326895, −2.76467468194080281123979967824, −1.84563578526410413114333585979, −0.34003975264294080460771492007, 1.09977730501083636858982056920, 2.05614080710086254309649094466, 3.60234893102084824972277382258, 4.07409279658638439388412752561, 5.14053366652607893350689472139, 5.86081145795725011627927541340, 6.43365920113222748795448380989, 7.35914414375337429022514719900, 8.160653279964057239563269486608, 8.850659536565065248780048757024

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