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Properties

Label 2-750-25.21-c1-0-14
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: $\chi_{750} (601, \cdot )$
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

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

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