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Properties

Label 2-750-75.17-c1-0-13
Degree $2$
Conductor $750$
Sign $-0.326 - 0.945i$
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  + (0.891 + 0.453i)2-s + (0.974 + 1.43i)3-s + (0.587 + 0.809i)4-s + (0.218 + 1.71i)6-s + (3.13 + 3.13i)7-s + (0.156 + 0.987i)8-s + (−1.10 + 2.79i)9-s + (−3.57 + 1.16i)11-s + (−0.585 + 1.63i)12-s + (−1.78 − 3.49i)13-s + (1.37 + 4.21i)14-s + (−0.309 + 0.951i)16-s + (−0.406 + 0.0644i)17-s + (−2.24 + 1.98i)18-s + (2.93 − 4.04i)19-s + ⋯
L(s)  = 1  + (0.630 + 0.321i)2-s + (0.562 + 0.826i)3-s + (0.293 + 0.404i)4-s + (0.0891 + 0.701i)6-s + (1.18 + 1.18i)7-s + (0.0553 + 0.349i)8-s + (−0.366 + 0.930i)9-s + (−1.07 + 0.350i)11-s + (−0.169 + 0.470i)12-s + (−0.494 − 0.969i)13-s + (0.366 + 1.12i)14-s + (−0.0772 + 0.237i)16-s + (−0.0986 + 0.0156i)17-s + (−0.529 + 0.468i)18-s + (0.674 − 0.928i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.326 - 0.945i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57446 + 2.20927i\)
\(L(\frac12)\) \(\approx\) \(1.57446 + 2.20927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.891 - 0.453i)T \)
3 \( 1 + (-0.974 - 1.43i)T \)
5 \( 1 \)
good7 \( 1 + (-3.13 - 3.13i)T + 7iT^{2} \)
11 \( 1 + (3.57 - 1.16i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.78 + 3.49i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.406 - 0.0644i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-2.93 + 4.04i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-2.28 + 4.48i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (1.49 - 1.08i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.69 - 1.96i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.70 + 2.39i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-3.24 - 1.05i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-3.71 + 3.71i)T - 43iT^{2} \)
47 \( 1 + (0.808 - 5.10i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (8.20 + 1.29i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-1.73 + 5.34i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.43 - 13.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.968 + 6.11i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (0.992 + 1.36i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.19 - 1.62i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-6.24 - 8.60i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.10 + 7.00i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (0.324 + 0.999i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (7.61 + 1.20i)T + (92.2 + 29.9i)T^{2} \)
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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

−10.76406646787493721837058807090, −9.695773888916157459670038277384, −8.742321345794480763285592087450, −8.067076811628144206339414874242, −7.38030969276982558674111081893, −5.75254519956555406208417677660, −5.06265104281094839908307354014, −4.57478801462530636782424529072, −2.93281077599732982176273096402, −2.38288426151010302519479643820, 1.15890539783248379240256945708, 2.24184964464190076155082560655, 3.50001206184675467983101945292, 4.51089909230444974198916611187, 5.51188026903631234742652154367, 6.69582773703478090068182024676, 7.69826958258194620027105407780, 7.915042613616085216486860468163, 9.302455580316179750839116890504, 10.21105473387886783631534226387

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