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Properties

Label 2-750-25.6-c1-0-0
Degree $2$
Conductor $750$
Sign $-0.637 - 0.770i$
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 − 0.381·7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.427 + 1.31i)11-s + (0.309 − 0.951i)12-s + (−0.763 + 2.35i)13-s + (0.118 + 0.363i)14-s + (0.309 − 0.951i)16-s + (−2.61 − 1.90i)17-s − 18-s + (−6.23 − 4.53i)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 − 0.144·7-s + (0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + (0.128 + 0.396i)11-s + (0.0892 − 0.274i)12-s + (−0.211 + 0.652i)13-s + (0.0315 + 0.0970i)14-s + (0.0772 − 0.237i)16-s + (−0.634 − 0.461i)17-s − 0.235·18-s + (−1.43 − 1.03i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.637 - 0.770i$
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: \((2,\ 750,\ (\ :1/2),\ -0.637 - 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102353 + 0.217513i\)
\(L(\frac12)\) \(\approx\) \(0.102353 + 0.217513i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.381T + 7T^{2} \)
11 \( 1 + (-0.427 - 1.31i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.763 - 2.35i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.61 + 1.90i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (6.23 + 4.53i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.38 - 4.25i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.381 + 0.277i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.54 + 2.57i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.47 - 7.60i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.38 - 7.33i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.70T + 43T^{2} \)
47 \( 1 + (9.47 - 6.88i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-7.35 + 5.34i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.427 - 1.31i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.23 - 6.88i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (8.47 + 6.15i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (11.7 - 8.50i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.85 + 11.8i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.73 + 1.98i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.16 - 5.20i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-3.85 - 11.8i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.54 - 3.30i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.73798146020142948003613852564, −9.800254405467695193165333102243, −9.232227778449076381629236977758, −8.345938777882195545530285378670, −7.08269279567952481709920279538, −6.36292566845250491742681534220, −4.93680552297490523383220404139, −4.35808501397310675397102211851, −3.07535923512870423348967913672, −1.76990449282016761812299874112, 0.13494919533082127167257836568, 1.93538992020680505622614215105, 3.64679019074251193477275012119, 4.81459077365901206722370254354, 5.82563596710610186366747127641, 6.48948709586712946515839770694, 7.33763301208382648498954336058, 8.382017006724669614779786351572, 8.861050230137794371130685073071, 10.28856283418562687054567636678

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