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Properties

Label 2-1520-380.379-c1-0-39
Degree $2$
Conductor $1520$
Sign $0.0537 + 0.998i$
Analytic cond. $12.1372$
Root an. cond. $3.48385$
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  − 1.10i·3-s + (0.602 − 2.15i)5-s − 2.26·7-s + 1.77·9-s + 1.56i·11-s + 6.54·13-s + (−2.38 − 0.665i)15-s − 2.59i·17-s + (3.27 + 2.87i)19-s + 2.50i·21-s + 3.39·23-s + (−4.27 − 2.59i)25-s − 5.28i·27-s − 0.621i·29-s + 7.20·31-s + ⋯
L(s)  = 1  − 0.638i·3-s + (0.269 − 0.963i)5-s − 0.856·7-s + 0.592·9-s + 0.471i·11-s + 1.81·13-s + (−0.614 − 0.171i)15-s − 0.628i·17-s + (0.751 + 0.659i)19-s + 0.546i·21-s + 0.707·23-s + (−0.854 − 0.518i)25-s − 1.01i·27-s − 0.115i·29-s + 1.29·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0537 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0537 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.0537 + 0.998i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.0537 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.867208714\)
\(L(\frac12)\) \(\approx\) \(1.867208714\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.602 + 2.15i)T \)
19 \( 1 + (-3.27 - 2.87i)T \)
good3 \( 1 + 1.10iT - 3T^{2} \)
7 \( 1 + 2.26T + 7T^{2} \)
11 \( 1 - 1.56iT - 11T^{2} \)
13 \( 1 - 6.54T + 13T^{2} \)
17 \( 1 + 2.59iT - 17T^{2} \)
23 \( 1 - 3.39T + 23T^{2} \)
29 \( 1 + 0.621iT - 29T^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 + 1.45T + 37T^{2} \)
41 \( 1 + 7.46iT - 41T^{2} \)
43 \( 1 + 5.63T + 43T^{2} \)
47 \( 1 + 3.82T + 47T^{2} \)
53 \( 1 - 5.96T + 53T^{2} \)
59 \( 1 + 8.28T + 59T^{2} \)
61 \( 1 + 2.70T + 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 + 7.20T + 71T^{2} \)
73 \( 1 - 0.644iT - 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 + 8.56T + 83T^{2} \)
89 \( 1 - 7.00iT - 89T^{2} \)
97 \( 1 - 5.28T + 97T^{2} \)
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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

−9.294257347440127311088383916991, −8.472309544561614831848819984203, −7.71006386042314262963009636053, −6.73585229747316191917361235760, −6.15870497456127983462135891242, −5.19279052639961583891962582613, −4.18107938601257307325186301296, −3.20894132689046229692758940896, −1.73942206217852778256860518330, −0.871188889111057589437183071083, 1.33856902228024375439641601795, 3.05611016451317402990094408653, 3.45048885310492380183884404872, 4.49647352301529515776823670151, 5.73296739192668629393178017577, 6.43112436213997678734052176464, 7.00582219846553372160915150567, 8.170307441383539313359135779647, 9.013977629339484538558259635286, 9.793210526085108865738570211430

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