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Properties

Label 2-750-25.6-c1-0-12
Degree $2$
Conductor $750$
Sign $0.998 - 0.0598i$
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 + 4.25·7-s + (−0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + (−1.51 − 4.64i)11-s + (−0.309 + 0.951i)12-s + (1.43 − 4.41i)13-s + (1.31 + 4.04i)14-s + (0.309 − 0.951i)16-s + (−0.815 − 0.592i)17-s + 18-s + (1 + 0.726i)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 + 1.60·7-s + (−0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + (−0.455 − 1.40i)11-s + (−0.0892 + 0.274i)12-s + (0.397 − 1.22i)13-s + (0.351 + 1.08i)14-s + (0.0772 − 0.237i)16-s + (−0.197 − 0.143i)17-s + 0.235·18-s + (0.229 + 0.166i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.998 - 0.0598i$
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: \((2,\ 750,\ (\ :1/2),\ 0.998 - 0.0598i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19565 + 0.0657785i\)
\(L(\frac12)\) \(\approx\) \(2.19565 + 0.0657785i\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.25T + 7T^{2} \)
11 \( 1 + (1.51 + 4.64i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.43 + 4.41i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.815 + 0.592i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1 - 0.726i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.38 - 4.25i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.42 - 2.48i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.826 + 0.600i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.31 - 10.1i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.42 + 4.37i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + (1.63 - 1.19i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-8.94 + 6.49i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.656 - 2.02i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.19 - 3.68i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-3.03 - 2.20i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (10.1 - 7.37i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.05 + 3.26i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.16 - 3.75i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.70 - 4.14i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.693 - 2.13i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (6.49 - 4.71i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.49521340320106976839057912108, −9.118252574148286100388138940876, −8.282547562311669614044059223485, −7.978635011584395462622243521462, −7.07177697743897394630394009090, −5.67008933724871363346992869766, −5.30065442139994235870696975589, −3.91531989000535899479227828384, −2.83829560991278952412545957196, −1.16700233627072570046156726286, 1.69275608064587737419343988181, 2.41482941594638487857711254986, 4.15924298205032581148403842246, 4.51852796997231100871714800423, 5.52972360693366475966029696852, 7.06698185184033691672505522191, 7.87158384940275701073006088456, 8.855388420267350770564874324426, 9.450436985348486015769344034327, 10.56416794198095108946358365305

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