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Properties

Label 2-300-15.2-c7-0-0
Degree $2$
Conductor $300$
Sign $-0.747 - 0.664i$
Analytic cond. $93.7155$
Root an. cond. $9.68067$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.05 − 46.0i)3-s + (565. + 565. i)7-s + (−2.05e3 − 741. i)9-s + 2.36e3i·11-s + (−732. + 732. i)13-s + (2.28e4 − 2.28e4i)17-s + 3.87e4i·19-s + (3.05e4 − 2.14e4i)21-s + (−2.39e4 − 2.39e4i)23-s + (−5.07e4 + 8.88e4i)27-s − 1.72e5·29-s − 1.19e5·31-s + (1.08e5 + 1.90e4i)33-s + (−9.36e4 − 9.36e4i)37-s + (2.78e4 + 3.96e4i)39-s + ⋯
L(s)  = 1  + (0.172 − 0.985i)3-s + (0.622 + 0.622i)7-s + (−0.940 − 0.339i)9-s + 0.534i·11-s + (−0.0924 + 0.0924i)13-s + (1.13 − 1.13i)17-s + 1.29i·19-s + (0.720 − 0.506i)21-s + (−0.409 − 0.409i)23-s + (−0.496 + 0.868i)27-s − 1.31·29-s − 0.717·31-s + (0.526 + 0.0921i)33-s + (−0.304 − 0.304i)37-s + (0.0751 + 0.106i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.747 - 0.664i$
Analytic conductor: \(93.7155\)
Root analytic conductor: \(9.68067\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :7/2),\ -0.747 - 0.664i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.07645600106\)
\(L(\frac12)\) \(\approx\) \(0.07645600106\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-8.05 + 46.0i)T \)
5 \( 1 \)
good7 \( 1 + (-565. - 565. i)T + 8.23e5iT^{2} \)
11 \( 1 - 2.36e3iT - 1.94e7T^{2} \)
13 \( 1 + (732. - 732. i)T - 6.27e7iT^{2} \)
17 \( 1 + (-2.28e4 + 2.28e4i)T - 4.10e8iT^{2} \)
19 \( 1 - 3.87e4iT - 8.93e8T^{2} \)
23 \( 1 + (2.39e4 + 2.39e4i)T + 3.40e9iT^{2} \)
29 \( 1 + 1.72e5T + 1.72e10T^{2} \)
31 \( 1 + 1.19e5T + 2.75e10T^{2} \)
37 \( 1 + (9.36e4 + 9.36e4i)T + 9.49e10iT^{2} \)
41 \( 1 + 6.73e5iT - 1.94e11T^{2} \)
43 \( 1 + (2.65e5 - 2.65e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (2.18e5 - 2.18e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (9.79e3 + 9.79e3i)T + 1.17e12iT^{2} \)
59 \( 1 + 1.54e6T + 2.48e12T^{2} \)
61 \( 1 + 3.94e5T + 3.14e12T^{2} \)
67 \( 1 + (8.98e5 + 8.98e5i)T + 6.06e12iT^{2} \)
71 \( 1 + 2.33e6iT - 9.09e12T^{2} \)
73 \( 1 + (2.64e6 - 2.64e6i)T - 1.10e13iT^{2} \)
79 \( 1 - 3.05e6iT - 1.92e13T^{2} \)
83 \( 1 + (2.34e6 + 2.34e6i)T + 2.71e13iT^{2} \)
89 \( 1 - 3.61e6T + 4.42e13T^{2} \)
97 \( 1 + (-3.89e6 - 3.89e6i)T + 8.07e13iT^{2} \)
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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

−11.09606847612911465450858577513, −9.814843250639681309849219159226, −8.865585163447244072644553291175, −7.84720442328520206773941661146, −7.26695315757080204522984342602, −5.95079945847412719467745377028, −5.16837065155928075906608335107, −3.55530818868598931907568868267, −2.25628341757853581966336987148, −1.43630260126463667640317845530, 0.01547527579046980177481480587, 1.52299343315089617296054060062, 3.09973657548156313432273753709, 4.00207338575058159892644877009, 5.04032512156708084951876454995, 5.98194768624407443284843201486, 7.49022981873843173064741865590, 8.315888233023831718808224441193, 9.288367922463537786946753633199, 10.23478938373414815270681416073

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