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Properties

Label 2-750-25.6-c1-0-9
Degree $2$
Conductor $750$
Sign $0.968 + 0.248i$
Analytic cond. $5.98878$
Root an. cond. $2.44719$
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  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s − 2·7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−1.61 − 4.97i)11-s + (0.309 − 0.951i)12-s + (1.5 − 4.61i)13-s + (−0.618 − 1.90i)14-s + (0.309 − 0.951i)16-s + (6.35 + 4.61i)17-s + 18-s + (2.23 + 1.62i)19-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.330 − 0.239i)6-s − 0.755·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.487 − 1.50i)11-s + (0.0892 − 0.274i)12-s + (0.416 − 1.28i)13-s + (−0.165 − 0.508i)14-s + (0.0772 − 0.237i)16-s + (1.54 + 1.11i)17-s + 0.235·18-s + (0.512 + 0.372i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.968 + 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05550 - 0.133341i\)
\(L(\frac12)\) \(\approx\) \(1.05550 - 0.133341i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 \)
good7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + (1.61 + 4.97i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.5 + 4.61i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-6.35 - 4.61i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.23 - 1.62i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.85 + 5.70i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.11 + 0.812i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3 + 2.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.663 + 2.04i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.88 - 5.79i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.23T + 43T^{2} \)
47 \( 1 + (-3.85 + 2.80i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-6.92 + 5.03i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.76 + 8.50i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.73 + 8.42i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.85 + 5.70i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-11.4 + 8.33i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.972 - 2.99i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.85 - 3.52i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.427 - 1.31i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (11.2 - 8.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.34964615143925021455170420694, −9.537805317535803251312117845399, −8.248338862293972004635718566090, −7.975497285941185969621854465718, −6.48248691457763743645028518979, −5.84128735457170805485733569146, −5.30454528423442238318683946454, −3.73619013067644831075704278417, −3.16045091177649475566511807503, −0.59181944214599629775821325079, 1.36972902705232675387277515565, 2.65698472824213349100530579633, 3.86944415383830531469308315959, 4.96928964668002958252492412578, 5.77752772553210582766396057707, 7.04094619460806537473878332181, 7.48286894287809848481502964738, 9.043106193474617439957447294321, 9.727211457675393396315111285626, 10.29515204559759286018591903872

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