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Properties

Label 2-75-5.2-c8-0-13
Degree $2$
Conductor $75$
Sign $0.991 + 0.130i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
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  + (7.34 + 7.34i)2-s + (33.0 − 33.0i)3-s − 148i·4-s + 486·6-s + (2.87e3 + 2.87e3i)7-s + (2.96e3 − 2.96e3i)8-s − 2.18e3i·9-s − 234·11-s + (−4.89e3 − 4.89e3i)12-s + (−1.19e4 + 1.19e4i)13-s + 4.22e4i·14-s + 5.74e3·16-s + (8.90e4 + 8.90e4i)17-s + (1.60e4 − 1.60e4i)18-s − 1.81e5i·19-s + ⋯
L(s)  = 1  + (0.459 + 0.459i)2-s + (0.408 − 0.408i)3-s − 0.578i·4-s + 0.375·6-s + (1.19 + 1.19i)7-s + (0.724 − 0.724i)8-s − 0.333i·9-s − 0.0159·11-s + (−0.236 − 0.236i)12-s + (−0.417 + 0.417i)13-s + 1.09i·14-s + 0.0876·16-s + (1.06 + 1.06i)17-s + (0.153 − 0.153i)18-s − 1.39i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.991 + 0.130i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.58949 - 0.235710i\)
\(L(\frac12)\) \(\approx\) \(3.58949 - 0.235710i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-33.0 + 33.0i)T \)
5 \( 1 \)
good2 \( 1 + (-7.34 - 7.34i)T + 256iT^{2} \)
7 \( 1 + (-2.87e3 - 2.87e3i)T + 5.76e6iT^{2} \)
11 \( 1 + 234T + 2.14e8T^{2} \)
13 \( 1 + (1.19e4 - 1.19e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (-8.90e4 - 8.90e4i)T + 6.97e9iT^{2} \)
19 \( 1 + 1.81e5iT - 1.69e10T^{2} \)
23 \( 1 + (-2.69e5 + 2.69e5i)T - 7.83e10iT^{2} \)
29 \( 1 - 2.40e5iT - 5.00e11T^{2} \)
31 \( 1 - 8.36e5T + 8.52e11T^{2} \)
37 \( 1 + (-6.08e5 - 6.08e5i)T + 3.51e12iT^{2} \)
41 \( 1 - 2.82e6T + 7.98e12T^{2} \)
43 \( 1 + (-2.80e6 + 2.80e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (5.39e6 + 5.39e6i)T + 2.38e13iT^{2} \)
53 \( 1 + (1.19e6 - 1.19e6i)T - 6.22e13iT^{2} \)
59 \( 1 - 1.27e7iT - 1.46e14T^{2} \)
61 \( 1 - 5.17e5T + 1.91e14T^{2} \)
67 \( 1 + (-2.06e6 - 2.06e6i)T + 4.06e14iT^{2} \)
71 \( 1 + 2.08e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.88e7 + 2.88e7i)T - 8.06e14iT^{2} \)
79 \( 1 - 4.21e7iT - 1.51e15T^{2} \)
83 \( 1 + (6.66e7 - 6.66e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 9.51e7iT - 3.93e15T^{2} \)
97 \( 1 + (-7.08e7 - 7.08e7i)T + 7.83e15iT^{2} \)
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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

−13.00815540857358151018021385746, −11.92804736490747005094767119657, −10.72161828626027114697407751191, −9.239533927244362622856128044234, −8.192067298856133163243548694877, −6.85865016063267525644476247027, −5.59472730288084429719634403542, −4.58886396829566901857450112384, −2.45555746967514716435869207084, −1.17152137032661375207576927699, 1.29830175434746065403103507239, 2.97034052074813939067322676274, 4.13445994986186759614583914155, 5.13125732194581415319086958823, 7.57360157801770434175578266151, 7.964454907992431261094182434449, 9.722953702814591783170907990108, 10.86959205211094297680423432050, 11.73319511083566414854556449219, 12.96544641591279369963551067093

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