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Properties

Label 2-750-125.89-c1-0-19
Degree $2$
Conductor $750$
Sign $-0.241 + 0.970i$
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  + (0.481 − 0.876i)2-s + (0.368 − 0.929i)3-s + (−0.535 − 0.844i)4-s + (0.106 − 2.23i)5-s + (−0.637 − 0.770i)6-s + (4.43 + 1.44i)7-s + (−0.998 + 0.0627i)8-s + (−0.728 − 0.684i)9-s + (−1.90 − 1.16i)10-s + (2.85 + 1.56i)11-s + (−0.982 + 0.187i)12-s + (3.06 − 3.25i)13-s + (3.40 − 3.19i)14-s + (−2.03 − 0.921i)15-s + (−0.425 + 0.904i)16-s + (6.23 + 3.95i)17-s + ⋯
L(s)  = 1  + (0.340 − 0.619i)2-s + (0.212 − 0.536i)3-s + (−0.267 − 0.422i)4-s + (0.0478 − 0.998i)5-s + (−0.260 − 0.314i)6-s + (1.67 + 0.544i)7-s + (−0.352 + 0.0221i)8-s + (−0.242 − 0.228i)9-s + (−0.602 − 0.369i)10-s + (0.860 + 0.472i)11-s + (−0.283 + 0.0540i)12-s + (0.848 − 0.903i)13-s + (0.908 − 0.853i)14-s + (−0.526 − 0.237i)15-s + (−0.106 + 0.226i)16-s + (1.51 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.241 + 0.970i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.241 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45076 - 1.85623i\)
\(L(\frac12)\) \(\approx\) \(1.45076 - 1.85623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.481 + 0.876i)T \)
3 \( 1 + (-0.368 + 0.929i)T \)
5 \( 1 + (-0.106 + 2.23i)T \)
good7 \( 1 + (-4.43 - 1.44i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (-2.85 - 1.56i)T + (5.89 + 9.28i)T^{2} \)
13 \( 1 + (-3.06 + 3.25i)T + (-0.816 - 12.9i)T^{2} \)
17 \( 1 + (-6.23 - 3.95i)T + (7.23 + 15.3i)T^{2} \)
19 \( 1 + (6.63 - 2.62i)T + (13.8 - 13.0i)T^{2} \)
23 \( 1 + (0.877 + 6.94i)T + (-22.2 + 5.71i)T^{2} \)
29 \( 1 + (0.463 - 0.119i)T + (25.4 - 13.9i)T^{2} \)
31 \( 1 + (5.18 - 8.16i)T + (-13.1 - 28.0i)T^{2} \)
37 \( 1 + (5.80 + 2.73i)T + (23.5 + 28.5i)T^{2} \)
41 \( 1 + (-2.10 - 0.265i)T + (39.7 + 10.1i)T^{2} \)
43 \( 1 + (4.36 - 6.00i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (6.74 + 0.424i)T + (46.6 + 5.89i)T^{2} \)
53 \( 1 + (0.503 + 0.416i)T + (9.93 + 52.0i)T^{2} \)
59 \( 1 + (0.579 + 3.03i)T + (-54.8 + 21.7i)T^{2} \)
61 \( 1 + (10.7 - 1.35i)T + (59.0 - 15.1i)T^{2} \)
67 \( 1 + (0.0477 - 0.185i)T + (-58.7 - 32.2i)T^{2} \)
71 \( 1 + (-0.304 + 4.84i)T + (-70.4 - 8.89i)T^{2} \)
73 \( 1 + (-8.30 - 1.58i)T + (67.8 + 26.8i)T^{2} \)
79 \( 1 + (-3.02 - 1.19i)T + (57.5 + 54.0i)T^{2} \)
83 \( 1 + (-1.82 - 4.60i)T + (-60.5 + 56.8i)T^{2} \)
89 \( 1 + (-1.30 + 6.84i)T + (-82.7 - 32.7i)T^{2} \)
97 \( 1 + (-3.57 - 13.9i)T + (-85.0 + 46.7i)T^{2} \)
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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

−10.35310162776573920347977166215, −9.027936158876359135051004939994, −8.340664184317718966412457982356, −8.000408426493320205632965734396, −6.32634038518359817178660023521, −5.47641938137941000577544150166, −4.61662944291275878968565347287, −3.60990146185409127689917958721, −1.86492587693400758247600483839, −1.34903742631419114480119903382, 1.79614294871630120843748425315, 3.48894347668502744203868289061, 4.13250036741448395529398072274, 5.20751909871844371363404085111, 6.18641039774253093332787991680, 7.20147167846089898148765800653, 7.87782006647014582915589195391, 8.777092664173221601270979674513, 9.655636696784068255005886965597, 10.79913401218460861757161983386

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