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Properties

Label 2-1500-15.2-c1-0-8
Degree $2$
Conductor $1500$
Sign $-0.263 - 0.964i$
Analytic cond. $11.9775$
Root an. cond. $3.46086$
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  + (1.50 + 0.858i)3-s + (−3.42 − 3.42i)7-s + (1.52 + 2.58i)9-s + 4.41i·11-s + (−2.37 + 2.37i)13-s + (2.76 − 2.76i)17-s + 5.73i·19-s + (−2.21 − 8.08i)21-s + (4.22 + 4.22i)23-s + (0.0829 + 5.19i)27-s − 9.67·29-s − 0.881·31-s + (−3.78 + 6.64i)33-s + (4.89 + 4.89i)37-s + (−5.61 + 1.53i)39-s + ⋯
L(s)  = 1  + (0.868 + 0.495i)3-s + (−1.29 − 1.29i)7-s + (0.509 + 0.860i)9-s + 1.33i·11-s + (−0.659 + 0.659i)13-s + (0.669 − 0.669i)17-s + 1.31i·19-s + (−0.483 − 1.76i)21-s + (0.880 + 0.880i)23-s + (0.0159 + 0.999i)27-s − 1.79·29-s − 0.158·31-s + (−0.659 + 1.15i)33-s + (0.804 + 0.804i)37-s + (−0.899 + 0.246i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.263 - 0.964i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.521872326\)
\(L(\frac12)\) \(\approx\) \(1.521872326\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.50 - 0.858i)T \)
5 \( 1 \)
good7 \( 1 + (3.42 + 3.42i)T + 7iT^{2} \)
11 \( 1 - 4.41iT - 11T^{2} \)
13 \( 1 + (2.37 - 2.37i)T - 13iT^{2} \)
17 \( 1 + (-2.76 + 2.76i)T - 17iT^{2} \)
19 \( 1 - 5.73iT - 19T^{2} \)
23 \( 1 + (-4.22 - 4.22i)T + 23iT^{2} \)
29 \( 1 + 9.67T + 29T^{2} \)
31 \( 1 + 0.881T + 31T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 - 4.74iT - 41T^{2} \)
43 \( 1 + (-2.35 + 2.35i)T - 43iT^{2} \)
47 \( 1 + (1.71 - 1.71i)T - 47iT^{2} \)
53 \( 1 + (0.0871 + 0.0871i)T + 53iT^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 8.20T + 61T^{2} \)
67 \( 1 + (-5.36 - 5.36i)T + 67iT^{2} \)
71 \( 1 + 5.14iT - 71T^{2} \)
73 \( 1 + (-0.0238 + 0.0238i)T - 73iT^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 + (3.11 + 3.11i)T + 83iT^{2} \)
89 \( 1 + 2.72T + 89T^{2} \)
97 \( 1 + (4.70 + 4.70i)T + 97iT^{2} \)
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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

−9.599951838469127479335121295317, −9.424478895982199303218285535979, −7.923662565648111870516450044844, −7.31003893374776062721843888148, −6.86293225487406602318071350322, −5.46322693954487900025757935054, −4.38110826627255933175613387680, −3.76799192395730756827849602634, −2.89388846647597606465914136685, −1.58363634427955265012318692075, 0.52519080506223747038349673517, 2.36142934446822446724400778896, 2.97753648433866053729823276855, 3.72385922135931779536015257882, 5.35513157826214923234917949485, 6.01812351268326462456103802446, 6.82478424474015504087602819820, 7.71992938717062131439634309729, 8.649270372885560344694487330606, 9.093232545947019852536004588896

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