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Properties

Label 2-750-125.91-c1-0-16
Degree $2$
Conductor $750$
Sign $0.203 + 0.979i$
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  + (0.728 + 0.684i)2-s + (−0.535 + 0.844i)3-s + (0.0627 + 0.998i)4-s + (−2.14 − 0.616i)5-s + (−0.968 + 0.248i)6-s + (−0.0420 − 0.129i)7-s + (−0.637 + 0.770i)8-s + (−0.425 − 0.904i)9-s + (−1.14 − 1.92i)10-s + (−2.93 − 2.75i)11-s + (−0.876 − 0.481i)12-s + (−0.623 − 1.32i)13-s + (0.0579 − 0.123i)14-s + (1.67 − 1.48i)15-s + (−0.992 + 0.125i)16-s + (0.339 − 5.39i)17-s + ⋯
L(s)  = 1  + (0.515 + 0.484i)2-s + (−0.309 + 0.487i)3-s + (0.0313 + 0.499i)4-s + (−0.961 − 0.275i)5-s + (−0.395 + 0.101i)6-s + (−0.0158 − 0.0489i)7-s + (−0.225 + 0.272i)8-s + (−0.141 − 0.301i)9-s + (−0.361 − 0.607i)10-s + (−0.884 − 0.830i)11-s + (−0.252 − 0.139i)12-s + (−0.173 − 0.367i)13-s + (0.0154 − 0.0328i)14-s + (0.431 − 0.383i)15-s + (−0.248 + 0.0313i)16-s + (0.0823 − 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.203 + 0.979i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.203 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.492850 - 0.401120i\)
\(L(\frac12)\) \(\approx\) \(0.492850 - 0.401120i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.728 - 0.684i)T \)
3 \( 1 + (0.535 - 0.844i)T \)
5 \( 1 + (2.14 + 0.616i)T \)
good7 \( 1 + (0.0420 + 0.129i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (2.93 + 2.75i)T + (0.690 + 10.9i)T^{2} \)
13 \( 1 + (0.623 + 1.32i)T + (-8.28 + 10.0i)T^{2} \)
17 \( 1 + (-0.339 + 5.39i)T + (-16.8 - 2.13i)T^{2} \)
19 \( 1 + (-0.344 - 0.542i)T + (-8.08 + 17.1i)T^{2} \)
23 \( 1 + (0.423 + 2.21i)T + (-21.3 + 8.46i)T^{2} \)
29 \( 1 + (-0.0190 + 0.00753i)T + (21.1 - 19.8i)T^{2} \)
31 \( 1 + (-0.0131 + 0.208i)T + (-30.7 - 3.88i)T^{2} \)
37 \( 1 + (9.37 - 1.18i)T + (35.8 - 9.20i)T^{2} \)
41 \( 1 + (-0.931 + 4.88i)T + (-38.1 - 15.0i)T^{2} \)
43 \( 1 + (2.57 - 1.86i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (2.66 + 3.21i)T + (-8.80 + 46.1i)T^{2} \)
53 \( 1 + (4.37 + 1.12i)T + (46.4 + 25.5i)T^{2} \)
59 \( 1 + (3.08 + 1.69i)T + (31.6 + 49.8i)T^{2} \)
61 \( 1 + (0.313 + 1.64i)T + (-56.7 + 22.4i)T^{2} \)
67 \( 1 + (11.0 + 4.38i)T + (48.8 + 45.8i)T^{2} \)
71 \( 1 + (0.746 + 0.902i)T + (-13.3 + 69.7i)T^{2} \)
73 \( 1 + (-6.68 + 3.67i)T + (39.1 - 61.6i)T^{2} \)
79 \( 1 + (1.68 - 2.65i)T + (-33.6 - 71.4i)T^{2} \)
83 \( 1 + (-6.85 - 10.8i)T + (-35.3 + 75.1i)T^{2} \)
89 \( 1 + (1.88 - 1.03i)T + (47.6 - 75.1i)T^{2} \)
97 \( 1 + (17.8 - 7.07i)T + (70.7 - 66.4i)T^{2} \)
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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

−10.34744660945036289877554433994, −9.147918380859287969310345313520, −8.306804100223044504281634830103, −7.57417615196597855921626213690, −6.64833646898963970800015468607, −5.38161786938584381544050061101, −4.92988274344886080070144226230, −3.75848655376507184057577149743, −2.90411605566394508498513583581, −0.27733965357415987719264921924, 1.71266916371968279196853095674, 2.98080645213524397058323302294, 4.11593113941042781833897071209, 5.00094798514871066200149382037, 6.08606574833558180205032357845, 7.09428684148788899730479939235, 7.77307245403743317481752920635, 8.753734607972652535856471589718, 10.04536231524519348844463967561, 10.69497379208922480025018436306

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