Properties

Label 2-1520-5.4-c1-0-23
Degree $2$
Conductor $1520$
Sign $0.948 + 0.316i$
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  − 0.414i·3-s + (−2.12 − 0.707i)5-s − 0.414i·7-s + 2.82·9-s + 1.41·11-s + 3.82i·13-s + (−0.292 + 0.878i)15-s + i·17-s − 19-s − 0.171·21-s − 3.24i·23-s + (3.99 + 3i)25-s − 2.41i·27-s + 1.82·29-s − 0.585·31-s + ⋯
L(s)  = 1  − 0.239i·3-s + (−0.948 − 0.316i)5-s − 0.156i·7-s + 0.942·9-s + 0.426·11-s + 1.06i·13-s + (−0.0756 + 0.226i)15-s + 0.242i·17-s − 0.229·19-s − 0.0374·21-s − 0.676i·23-s + (0.799 + 0.600i)25-s − 0.464i·27-s + 0.339·29-s − 0.105·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(12.1372\)
Root analytic conductor: \(3.48385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :1/2),\ 0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.532713202\)
\(L(\frac12)\) \(\approx\) \(1.532713202\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 + 0.707i)T \)
19 \( 1 + T \)
good3 \( 1 + 0.414iT - 3T^{2} \)
7 \( 1 + 0.414iT - 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 - 3.82iT - 13T^{2} \)
17 \( 1 - iT - 17T^{2} \)
23 \( 1 + 3.24iT - 23T^{2} \)
29 \( 1 - 1.82T + 29T^{2} \)
31 \( 1 + 0.585T + 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 6.24iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 3.82iT - 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 - 0.585T + 61T^{2} \)
67 \( 1 + 8.07iT - 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + 8.17iT - 73T^{2} \)
79 \( 1 - 4.82T + 79T^{2} \)
83 \( 1 - 14.4iT - 83T^{2} \)
89 \( 1 - 12.2T + 89T^{2} \)
97 \( 1 + 3.65iT - 97T^{2} \)
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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

−9.331850670429909314730606664207, −8.562293288078245985176775632670, −7.84560106022582521904229303240, −6.90747819312700299355169184692, −6.53847824451944078476942820375, −5.06687546685037456833481733925, −4.25801468238873508664717172334, −3.67564156476820274354921270460, −2.12888077900738100818390995859, −0.900047003055115815618466990919, 0.916294492091942959435409000840, 2.55503064524296876872992391507, 3.65674990583320570145715394290, 4.27739640598052615362890623464, 5.30627092516066515155412615147, 6.31017275991180559942611282348, 7.34367436114907097030338167731, 7.69884805557043747764008717463, 8.726906594499942790562105066266, 9.520050946056400965486895926903

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