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Properties

Label 2-300-3.2-c8-0-37
Degree $2$
Conductor $300$
Sign $0.288 + 0.957i$
Analytic cond. $122.213$
Root an. cond. $11.0550$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (23.4 + 77.5i)3-s + 256.·7-s + (−5.46e3 + 3.62e3i)9-s − 1.24e4i·11-s + 3.38e3·13-s + 5.09e4i·17-s + 4.63e4·19-s + (6.00e3 + 1.99e4i)21-s + 3.19e5i·23-s + (−4.09e5 − 3.38e5i)27-s − 3.10e5i·29-s − 8.34e5·31-s + (9.66e5 − 2.91e5i)33-s − 2.88e6·37-s + (7.91e4 + 2.62e5i)39-s + ⋯
L(s)  = 1  + (0.288 + 0.957i)3-s + 0.106·7-s + (−0.833 + 0.553i)9-s − 0.851i·11-s + 0.118·13-s + 0.609i·17-s + 0.355·19-s + (0.0309 + 0.102i)21-s + 1.14i·23-s + (−0.770 − 0.637i)27-s − 0.439i·29-s − 0.904·31-s + (0.815 − 0.245i)33-s − 1.53·37-s + (0.0342 + 0.113i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.288 + 0.957i$
Analytic conductor: \(122.213\)
Root analytic conductor: \(11.0550\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :4),\ 0.288 + 0.957i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.9648607964\)
\(L(\frac12)\) \(\approx\) \(0.9648607964\)
\(L(5)\) 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 \( 1 + (-23.4 - 77.5i)T \)
5 \( 1 \)
good7 \( 1 - 256.T + 5.76e6T^{2} \)
11 \( 1 + 1.24e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.38e3T + 8.15e8T^{2} \)
17 \( 1 - 5.09e4iT - 6.97e9T^{2} \)
19 \( 1 - 4.63e4T + 1.69e10T^{2} \)
23 \( 1 - 3.19e5iT - 7.83e10T^{2} \)
29 \( 1 + 3.10e5iT - 5.00e11T^{2} \)
31 \( 1 + 8.34e5T + 8.52e11T^{2} \)
37 \( 1 + 2.88e6T + 3.51e12T^{2} \)
41 \( 1 + 2.24e6iT - 7.98e12T^{2} \)
43 \( 1 - 5.12e6T + 1.16e13T^{2} \)
47 \( 1 + 3.54e6iT - 2.38e13T^{2} \)
53 \( 1 - 5.57e5iT - 6.22e13T^{2} \)
59 \( 1 + 1.90e7iT - 1.46e14T^{2} \)
61 \( 1 + 4.81e6T + 1.91e14T^{2} \)
67 \( 1 - 1.45e7T + 4.06e14T^{2} \)
71 \( 1 - 3.50e6iT - 6.45e14T^{2} \)
73 \( 1 + 4.64e7T + 8.06e14T^{2} \)
79 \( 1 + 4.67e7T + 1.51e15T^{2} \)
83 \( 1 + 1.10e7iT - 2.25e15T^{2} \)
89 \( 1 + 7.11e7iT - 3.93e15T^{2} \)
97 \( 1 - 6.59e7T + 7.83e15T^{2} \)
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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

−10.16677146600087821533137527686, −9.184716421197024638532278715470, −8.465689024829210079046640220201, −7.44390957158358812265306191132, −5.96204978580858544214017579516, −5.19681270309918612682283819615, −3.91416036437723639029263369683, −3.19631592156232923314435224074, −1.79043983622268165341310894276, −0.19528426857499770748336094766, 1.06673246752448637508787357693, 2.11129710812072326077378791309, 3.14761923446860049172964594542, 4.56702445048603176892232096536, 5.77040113828926102389231028985, 6.89461172147329263837470455184, 7.51689296919557591531962670164, 8.591210383261534769636136001142, 9.414411441072283588038098865481, 10.58985354093940614931333189870

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