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Properties

Label 2-1520-1520.189-c0-0-6
Degree $2$
Conductor $1520$
Sign $0.923 + 0.382i$
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  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s − i·5-s − 1.00·6-s − 7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)10-s + (0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + (0.707 − 0.707i)14-s + (0.707 − 0.707i)15-s − 1.00·16-s − i·17-s + (0.707 − 0.707i)19-s − 1.00·20-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s − i·5-s − 1.00·6-s − 7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)10-s + (0.707 − 0.707i)12-s + (−0.707 − 0.707i)13-s + (0.707 − 0.707i)14-s + (0.707 − 0.707i)15-s − 1.00·16-s − i·17-s + (0.707 − 0.707i)19-s − 1.00·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7835125603\)
\(L(\frac12)\) \(\approx\) \(0.7835125603\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
17 \( 1 + iT - T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
59 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−9.460674522501369346209944625092, −9.047070660660268335780254105304, −8.184844214934337419255514428214, −7.32261604590078235452069942766, −6.55685334947710657358360249589, −5.31502011100506187701412979596, −4.91095372551600153974794357312, −3.61128416579616861925247474521, −2.58849494544677739444930765225, −0.73608104669461983488924517008, 1.66824405569572597715882493225, 2.62024498961102434629209962522, 3.24449585283641692459417404995, 4.22291508916034734299642246034, 5.94036302323436948812801378120, 6.88074002683771779680644685180, 7.44158713749325218652908747655, 8.019463094750664397749578840028, 9.147114225966245265821650614149, 9.566926179539050697063234793515

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