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Properties

Label 2-750-125.104-c1-0-15
Degree $2$
Conductor $750$
Sign $-0.0490 + 0.998i$
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  + (−0.248 − 0.968i)2-s + (0.982 − 0.187i)3-s + (−0.876 + 0.481i)4-s + (−2.12 + 0.709i)5-s + (−0.425 − 0.904i)6-s + (−0.931 + 1.28i)7-s + (0.684 + 0.728i)8-s + (0.929 − 0.368i)9-s + (1.21 + 1.87i)10-s + (2.53 − 0.649i)11-s + (−0.770 + 0.637i)12-s + (−1.61 − 4.08i)13-s + (1.47 + 0.583i)14-s + (−1.94 + 1.09i)15-s + (0.535 − 0.844i)16-s + (1.83 − 3.34i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.684i)2-s + (0.567 − 0.108i)3-s + (−0.438 + 0.240i)4-s + (−0.948 + 0.317i)5-s + (−0.173 − 0.369i)6-s + (−0.351 + 0.484i)7-s + (0.242 + 0.257i)8-s + (0.309 − 0.122i)9-s + (0.384 + 0.593i)10-s + (0.763 − 0.195i)11-s + (−0.222 + 0.184i)12-s + (−0.448 − 1.13i)13-s + (0.393 + 0.155i)14-s + (−0.503 + 0.282i)15-s + (0.133 − 0.211i)16-s + (0.446 − 0.811i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0490 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.0490 + 0.998i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.0490 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874063 - 0.918081i\)
\(L(\frac12)\) \(\approx\) \(0.874063 - 0.918081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.248 + 0.968i)T \)
3 \( 1 + (-0.982 + 0.187i)T \)
5 \( 1 + (2.12 - 0.709i)T \)
good7 \( 1 + (0.931 - 1.28i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-2.53 + 0.649i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (1.61 + 4.08i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (-1.83 + 3.34i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (-0.853 + 4.47i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (-5.35 - 0.337i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (-8.62 - 1.08i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (3.14 + 1.73i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (9.30 + 5.90i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (0.679 + 10.8i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (-9.10 + 2.95i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (7.47 - 7.96i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (-6.76 - 3.18i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (-2.49 - 3.02i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (0.581 - 9.24i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-1.07 - 8.49i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-4.81 - 4.51i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (11.4 + 9.49i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (3.27 + 17.1i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (11.7 + 2.24i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (-4.33 + 5.23i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (1.31 - 10.4i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.23958183544566308825762899385, −9.023894938584127760153380451095, −8.800407232332889284625254817921, −7.51094279127745240280960012654, −7.04180539995207662459279497387, −5.50627253087115766380539061638, −4.36543232076825620407659616847, −3.20454311679780269736213619788, −2.71228603249950219007878560451, −0.74169046289611950324189483513, 1.37052650670231073235912708837, 3.38345787245729996290777969927, 4.15525613372790688017691386569, 5.03179440654752404844458099201, 6.59719359958628698865800823123, 7.04494047392804992295065803263, 8.118387981395930403934141479323, 8.637558143184152694630881266436, 9.592714265549978951564724851878, 10.26753548656682686407618829668

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