Properties

Label 2-750-125.104-c1-0-2
Degree $2$
Conductor $750$
Sign $-0.194 - 0.980i$
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 + (1.78 + 1.34i)5-s + (0.425 + 0.904i)6-s + (−1.01 + 1.39i)7-s + (0.684 + 0.728i)8-s + (0.929 − 0.368i)9-s + (0.853 − 2.06i)10-s + (−2.53 + 0.650i)11-s + (0.770 − 0.637i)12-s + (−0.512 − 1.29i)13-s + (1.59 + 0.632i)14-s + (−2.00 − 0.981i)15-s + (0.535 − 0.844i)16-s + (0.822 − 1.49i)17-s + ⋯
L(s)  = 1  + (−0.175 − 0.684i)2-s + (−0.567 + 0.108i)3-s + (−0.438 + 0.240i)4-s + (0.800 + 0.599i)5-s + (0.173 + 0.369i)6-s + (−0.381 + 0.525i)7-s + (0.242 + 0.257i)8-s + (0.309 − 0.122i)9-s + (0.269 − 0.653i)10-s + (−0.764 + 0.196i)11-s + (0.222 − 0.184i)12-s + (−0.142 − 0.359i)13-s + (0.427 + 0.169i)14-s + (−0.518 − 0.253i)15-s + (0.133 − 0.211i)16-s + (0.199 − 0.362i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.194 - 0.980i$
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: \((2,\ 750,\ (\ :1/2),\ -0.194 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.395093 + 0.481004i\)
\(L(\frac12)\) \(\approx\) \(0.395093 + 0.481004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 - 1.34i)T \)
good7 \( 1 + (1.01 - 1.39i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.53 - 0.650i)T + (9.63 - 5.29i)T^{2} \)
13 \( 1 + (0.512 + 1.29i)T + (-9.47 + 8.89i)T^{2} \)
17 \( 1 + (-0.822 + 1.49i)T + (-9.10 - 14.3i)T^{2} \)
19 \( 1 + (1.34 - 7.02i)T + (-17.6 - 6.99i)T^{2} \)
23 \( 1 + (2.71 + 0.170i)T + (22.8 + 2.88i)T^{2} \)
29 \( 1 + (4.59 + 0.580i)T + (28.0 + 7.21i)T^{2} \)
31 \( 1 + (8.33 + 4.58i)T + (16.6 + 26.1i)T^{2} \)
37 \( 1 + (6.93 + 4.40i)T + (15.7 + 33.4i)T^{2} \)
41 \( 1 + (-0.785 - 12.4i)T + (-40.6 + 5.13i)T^{2} \)
43 \( 1 + (-0.760 + 0.246i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (3.97 - 4.23i)T + (-2.95 - 46.9i)T^{2} \)
53 \( 1 + (-12.8 - 6.06i)T + (33.7 + 40.8i)T^{2} \)
59 \( 1 + (-9.18 - 11.1i)T + (-11.0 + 57.9i)T^{2} \)
61 \( 1 + (-0.492 + 7.82i)T + (-60.5 - 7.64i)T^{2} \)
67 \( 1 + (-0.411 - 3.25i)T + (-64.8 + 16.6i)T^{2} \)
71 \( 1 + (-1.68 - 1.58i)T + (4.45 + 70.8i)T^{2} \)
73 \( 1 + (10.3 + 8.57i)T + (13.6 + 71.7i)T^{2} \)
79 \( 1 + (-1.94 - 10.2i)T + (-73.4 + 29.0i)T^{2} \)
83 \( 1 + (-2.90 - 0.554i)T + (77.1 + 30.5i)T^{2} \)
89 \( 1 + (7.31 - 8.84i)T + (-16.6 - 87.4i)T^{2} \)
97 \( 1 + (-0.598 + 4.73i)T + (-93.9 - 24.1i)T^{2} \)
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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

−10.52092557796458523736721534349, −9.905740865118339991415444160712, −9.299816944085500128098564553013, −8.054289847245818836087157123744, −7.13820525827407196936189445586, −5.80359875822076535216200695021, −5.53775398183242289836891259227, −4.00856035184261592844325305671, −2.84688610929030217488905522346, −1.80727690964309017436385198730, 0.34482456151694646242315259717, 2.01005905264321089579778046638, 3.83636279891943886379780463779, 5.11951839150336028031087320939, 5.53059658631088989660861169391, 6.72275599261787454408740568533, 7.21916150181173695301940922280, 8.510500625672251613806663151437, 9.139744750327516025037310419489, 10.16548351823303526216818260219

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