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Properties

Label 2-1500-25.6-c1-0-2
Degree $2$
Conductor $1500$
Sign $-0.577 - 0.816i$
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  + (0.809 − 0.587i)3-s − 3.80·7-s + (0.309 − 0.951i)9-s + (0.0589 + 0.181i)11-s + (−0.518 + 1.59i)13-s + (−3.72 − 2.70i)17-s + (2.13 + 1.55i)19-s + (−3.08 + 2.23i)21-s + (1.96 + 6.04i)23-s + (−0.309 − 0.951i)27-s + (−2.03 + 1.48i)29-s + (−3.03 − 2.20i)31-s + (0.154 + 0.112i)33-s + (−3.66 + 11.2i)37-s + (0.518 + 1.59i)39-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s − 1.44·7-s + (0.103 − 0.317i)9-s + (0.0177 + 0.0546i)11-s + (−0.143 + 0.442i)13-s + (−0.903 − 0.656i)17-s + (0.490 + 0.356i)19-s + (−0.672 + 0.488i)21-s + (0.409 + 1.26i)23-s + (−0.0594 − 0.183i)27-s + (−0.378 + 0.275i)29-s + (−0.544 − 0.395i)31-s + (0.0268 + 0.0195i)33-s + (−0.602 + 1.85i)37-s + (0.0831 + 0.255i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5458319585\)
\(L(\frac12)\) \(\approx\) \(0.5458319585\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 \)
good7 \( 1 + 3.80T + 7T^{2} \)
11 \( 1 + (-0.0589 - 0.181i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.518 - 1.59i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.72 + 2.70i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.13 - 1.55i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.96 - 6.04i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.03 - 1.48i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.03 + 2.20i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.66 - 11.2i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.22 - 6.83i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.22T + 43T^{2} \)
47 \( 1 + (-3.67 + 2.67i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.62 - 5.54i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.20 + 6.79i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.94 - 9.06i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (4.89 + 3.55i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (10.7 - 7.81i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.61 + 4.95i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.51 + 1.82i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.78 + 2.74i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.30 - 13.2i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.41 - 3.93i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.553500267397503938444431291155, −9.167933874937217942236747558072, −8.183196520335010237998885709014, −7.14974797097690376473194498816, −6.74257604495741640512191789004, −5.81024581167389977855145563685, −4.70222938611902174095070595250, −3.49867228751030588422018117578, −2.92091731596506278600594980012, −1.58640590925456778485039172552, 0.19133172594630503542053706750, 2.17029091730761120888628608552, 3.17502158007252595540260299777, 3.88006435496325878062695584602, 4.98105681588877574595327150329, 6.00661441344696902315320321950, 6.79084722162108354901538976883, 7.53369893025300285131409367721, 8.769657540818285922054055505796, 9.022669890112108699940357761298

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