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Properties

Label 2-300-5.2-c10-0-2
Degree $2$
Conductor $300$
Sign $-0.945 - 0.326i$
Analytic cond. $190.607$
Root an. cond. $13.8060$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−99.2 + 99.2i)3-s + (6.76e3 + 6.76e3i)7-s − 1.96e4i·9-s − 2.30e4·11-s + (4.17e5 − 4.17e5i)13-s + (−1.39e6 − 1.39e6i)17-s − 4.30e6i·19-s − 1.34e6·21-s + (−7.67e6 + 7.67e6i)23-s + (1.95e6 + 1.95e6i)27-s + 3.32e6i·29-s − 1.16e7·31-s + (2.28e6 − 2.28e6i)33-s + (8.35e7 + 8.35e7i)37-s + 8.27e7i·39-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.402 + 0.402i)7-s − 0.333i·9-s − 0.143·11-s + (1.12 − 1.12i)13-s + (−0.983 − 0.983i)17-s − 1.74i·19-s − 0.328·21-s + (−1.19 + 1.19i)23-s + (0.136 + 0.136i)27-s + 0.162i·29-s − 0.406·31-s + (0.0584 − 0.0584i)33-s + (1.20 + 1.20i)37-s + 0.917i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.945 - 0.326i$
Analytic conductor: \(190.607\)
Root analytic conductor: \(13.8060\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :5),\ -0.945 - 0.326i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.2219883469\)
\(L(\frac12)\) \(\approx\) \(0.2219883469\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (99.2 - 99.2i)T \)
5 \( 1 \)
good7 \( 1 + (-6.76e3 - 6.76e3i)T + 2.82e8iT^{2} \)
11 \( 1 + 2.30e4T + 2.59e10T^{2} \)
13 \( 1 + (-4.17e5 + 4.17e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (1.39e6 + 1.39e6i)T + 2.01e12iT^{2} \)
19 \( 1 + 4.30e6iT - 6.13e12T^{2} \)
23 \( 1 + (7.67e6 - 7.67e6i)T - 4.14e13iT^{2} \)
29 \( 1 - 3.32e6iT - 4.20e14T^{2} \)
31 \( 1 + 1.16e7T + 8.19e14T^{2} \)
37 \( 1 + (-8.35e7 - 8.35e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 1.04e8T + 1.34e16T^{2} \)
43 \( 1 + (-1.74e8 + 1.74e8i)T - 2.16e16iT^{2} \)
47 \( 1 + (1.83e8 + 1.83e8i)T + 5.25e16iT^{2} \)
53 \( 1 + (4.56e8 - 4.56e8i)T - 1.74e17iT^{2} \)
59 \( 1 - 9.95e8iT - 5.11e17T^{2} \)
61 \( 1 + 2.16e8T + 7.13e17T^{2} \)
67 \( 1 + (2.33e8 + 2.33e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 6.18e8T + 3.25e18T^{2} \)
73 \( 1 + (-7.26e8 + 7.26e8i)T - 4.29e18iT^{2} \)
79 \( 1 - 1.14e9iT - 9.46e18T^{2} \)
83 \( 1 + (-4.89e9 + 4.89e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 8.94e9iT - 3.11e19T^{2} \)
97 \( 1 + (-3.32e9 - 3.32e9i)T + 7.37e19iT^{2} \)
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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

−10.58077770426577981989457871775, −9.443588958927935460737868449109, −8.672939486291070476260832904260, −7.62113425885290938501815318784, −6.45527347455209196748159922861, −5.47644660813558879814793636221, −4.69461415488527028568124750702, −3.46741279813007808240186750431, −2.37029168101362365992517989553, −0.970191627561600650688329368094, 0.04735510250533229522833412561, 1.40210376971215513004586222796, 2.05549231876022029909641446387, 3.81497245952486293380540848654, 4.48666519755943210409370596649, 6.05047081539461250256155940716, 6.41838645502537540798998182898, 7.81748593054938420374141690049, 8.412915933370645209946960973376, 9.650043489416405794471990027968

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