Properties

Label 2-750-75.17-c1-0-27
Degree $2$
Conductor $750$
Sign $-0.764 + 0.644i$
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 + (1.37 − 1.05i)3-s + (0.587 + 0.809i)4-s + (−1.70 + 0.313i)6-s + (−0.462 − 0.462i)7-s + (−0.156 − 0.987i)8-s + (0.784 − 2.89i)9-s + (−2.73 + 0.888i)11-s + (1.66 + 0.494i)12-s + (−1.97 − 3.86i)13-s + (0.202 + 0.621i)14-s + (−0.309 + 0.951i)16-s + (−5.75 + 0.911i)17-s + (−2.01 + 2.22i)18-s + (4.28 − 5.89i)19-s + ⋯
L(s)  = 1  + (−0.630 − 0.321i)2-s + (0.794 − 0.607i)3-s + (0.293 + 0.404i)4-s + (−0.695 + 0.127i)6-s + (−0.174 − 0.174i)7-s + (−0.0553 − 0.349i)8-s + (0.261 − 0.965i)9-s + (−0.824 + 0.267i)11-s + (0.479 + 0.142i)12-s + (−0.546 − 1.07i)13-s + (0.0539 + 0.166i)14-s + (−0.0772 + 0.237i)16-s + (−1.39 + 0.221i)17-s + (−0.474 + 0.524i)18-s + (0.982 − 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.764 + 0.644i$
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: \((2,\ 750,\ (\ :1/2),\ -0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360441 - 0.986812i\)
\(L(\frac12)\) \(\approx\) \(0.360441 - 0.986812i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 + 1.05i)T \)
5 \( 1 \)
good7 \( 1 + (0.462 + 0.462i)T + 7iT^{2} \)
11 \( 1 + (2.73 - 0.888i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.97 + 3.86i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (5.75 - 0.911i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-4.28 + 5.89i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.316 - 0.622i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-2.60 + 1.89i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.82 + 3.50i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.93 + 0.988i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-6.34 - 2.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-5.08 + 5.08i)T - 43iT^{2} \)
47 \( 1 + (-0.474 + 2.99i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (2.23 + 0.353i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (2.66 - 8.19i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.77 - 8.55i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-2.22 - 14.0i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (7.15 + 9.84i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.498 + 0.254i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-6.00 - 8.25i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.742 - 4.68i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (1.78 + 5.49i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.26 + 0.833i)T + (92.2 + 29.9i)T^{2} \)
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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

−9.876531175113624703713645231928, −9.147319285241937024405025075430, −8.351920432489021804484866743902, −7.44913445370403029552222088329, −7.02298568470625345275231401938, −5.67427579069848191707819310203, −4.29230037183459673320179926966, −2.93795195316070297055123483002, −2.28914321766033309924767072013, −0.56688129613311744025731701543, 1.93843205916023114214474648380, 3.00904029513498701286529093647, 4.33088801079180814976713953468, 5.27757107618635743636916960402, 6.46099629666394268715858622261, 7.52069616848823080268380784292, 8.135570481787751382632718831218, 9.228936692329950720274999357053, 9.450205416125788188940684931167, 10.54920571887015994974046392631

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