Properties

Label 2-1520-95.94-c0-0-0
Degree $2$
Conductor $1520$
Sign $-0.5 - 0.866i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + 1.73i·7-s − 9-s − 11-s + 1.73i·17-s − 19-s + (−0.499 + 0.866i)25-s + (1.49 − 0.866i)35-s + 1.73i·43-s + (0.5 + 0.866i)45-s − 1.73i·47-s − 1.99·49-s + (0.5 + 0.866i)55-s + 61-s − 1.73i·63-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + 1.73i·7-s − 9-s − 11-s + 1.73i·17-s − 19-s + (−0.499 + 0.866i)25-s + (1.49 − 0.866i)35-s + 1.73i·43-s + (0.5 + 0.866i)45-s − 1.73i·47-s − 1.99·49-s + (0.5 + 0.866i)55-s + 61-s − 1.73i·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.5 - 0.866i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (1329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ -0.5 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5028177076\)
\(L(\frac12)\) \(\approx\) \(0.5028177076\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + T \)
good3 \( 1 + T^{2} \)
7 \( 1 - 1.73iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 + 1.73iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.782557261364068352827192366437, −8.807647450690743464868494463155, −8.397181526268815945055729163598, −8.008545052048966068933340175828, −6.45890558918095272640691575426, −5.65315880534468004626671256238, −5.20705805303697103068349515940, −4.04924237265337787017905063337, −2.85506453432692718727255649150, −1.94070536686646990624952468354, 0.37341246577719853444552760075, 2.48255188547650921870592578027, 3.29773739591657406364167793052, 4.24643962016331608623978551737, 5.15608891542934298169800485125, 6.32670970948055180215858782932, 7.15206240176779630013688111262, 7.62491186303668851065249892190, 8.403682581522614190530942192175, 9.531375965645854006734482836514

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