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Properties

Label 2-750-25.6-c1-0-6
Degree $2$
Conductor $750$
Sign $-0.0827 - 0.996i$
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 + 3.23·7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (1.63 + 5.04i)11-s + (0.309 − 0.951i)12-s + (1.82 − 5.61i)13-s + (0.998 + 3.07i)14-s + (0.309 − 0.951i)16-s + (−0.828 − 0.602i)17-s + 18-s + (3.64 + 2.64i)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 + 1.22·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (0.493 + 1.51i)11-s + (0.0892 − 0.274i)12-s + (0.506 − 1.55i)13-s + (0.266 + 0.821i)14-s + (0.0772 − 0.237i)16-s + (−0.201 − 0.146i)17-s + 0.235·18-s + (0.836 + 0.607i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0827 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0827 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.0827 - 0.996i$
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: \((2,\ 750,\ (\ :1/2),\ -0.0827 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11302 + 1.20927i\)
\(L(\frac12)\) \(\approx\) \(1.11302 + 1.20927i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.23T + 7T^{2} \)
11 \( 1 + (-1.63 - 5.04i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.82 + 5.61i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.828 + 0.602i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.64 - 2.64i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.595 - 1.83i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.210 + 0.153i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.81 - 1.31i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.95 - 9.08i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.07 - 9.47i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.30T + 43T^{2} \)
47 \( 1 + (-8.51 + 6.18i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.86 + 1.35i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.00 - 6.17i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.21 + 3.75i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.51 + 5.46i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (6.90 - 5.01i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.22 - 3.76i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.8 + 7.85i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.05 + 4.40i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.226 + 0.697i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-13.9 + 10.1i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.40121722872234806804112457371, −9.864014322743995024914876125522, −8.682975299735127555348726149525, −7.87705796091985465398055280114, −7.16153761891907610912537066407, −6.04753996314550559076576728312, −5.09529089572717323260215321143, −4.58825071202373618330123019884, −3.31738257453982973741902817486, −1.43858611463357744189044653490, 1.00343103313018371868135687319, 2.11197232851903068212733802159, 3.64806610717157002229943132042, 4.60366894926453208818118716894, 5.56969625727729004232053160352, 6.46251226238783441559128885518, 7.53247637822329003551619040111, 8.732096690202420990802956368291, 9.057930694336913838993519316413, 10.54789238128203730835490540125

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