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Properties

Label 2-5-5.4-c7-0-1
Degree $2$
Conductor $5$
Sign $0.268 + 0.963i$
Analytic cond. $1.56192$
Root an. cond. $1.24977$
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  − 10.7i·2-s − 32.3i·3-s + 12.0·4-s + (75 + 269. i)5-s − 347.·6-s + 420. i·7-s − 1.50e3i·8-s + 1.14e3·9-s + (2.90e3 − 807. i)10-s − 6.82e3·11-s − 387. i·12-s + 1.01e4i·13-s + 4.52e3·14-s + (8.69e3 − 2.42e3i)15-s − 1.47e4·16-s − 1.56e4i·17-s + ⋯
L(s)  = 1  − 0.951i·2-s − 0.690i·3-s + 0.0937·4-s + (0.268 + 0.963i)5-s − 0.657·6-s + 0.462i·7-s − 1.04i·8-s + 0.522·9-s + (0.917 − 0.255i)10-s − 1.54·11-s − 0.0647i·12-s + 1.28i·13-s + 0.440·14-s + (0.665 − 0.185i)15-s − 0.897·16-s − 0.774i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(1.56192\)
Root analytic conductor: \(1.24977\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :7/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.00927 - 0.766572i\)
\(L(\frac12)\) \(\approx\) \(1.00927 - 0.766572i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-75 - 269. i)T \)
good2 \( 1 + 10.7iT - 128T^{2} \)
3 \( 1 + 32.3iT - 2.18e3T^{2} \)
7 \( 1 - 420. iT - 8.23e5T^{2} \)
11 \( 1 + 6.82e3T + 1.94e7T^{2} \)
13 \( 1 - 1.01e4iT - 6.27e7T^{2} \)
17 \( 1 + 1.56e4iT - 4.10e8T^{2} \)
19 \( 1 - 6.86e3T + 8.93e8T^{2} \)
23 \( 1 + 2.92e4iT - 3.40e9T^{2} \)
29 \( 1 - 2.55e4T + 1.72e10T^{2} \)
31 \( 1 - 8.21e4T + 2.75e10T^{2} \)
37 \( 1 - 2.23e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.33e5T + 1.94e11T^{2} \)
43 \( 1 + 7.08e5iT - 2.71e11T^{2} \)
47 \( 1 + 5.82e3iT - 5.06e11T^{2} \)
53 \( 1 - 5.89e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.43e6T + 2.48e12T^{2} \)
61 \( 1 - 1.38e6T + 3.14e12T^{2} \)
67 \( 1 - 2.71e6iT - 6.06e12T^{2} \)
71 \( 1 + 4.81e5T + 9.09e12T^{2} \)
73 \( 1 + 1.48e6iT - 1.10e13T^{2} \)
79 \( 1 + 1.05e6T + 1.92e13T^{2} \)
83 \( 1 - 2.60e6iT - 2.71e13T^{2} \)
89 \( 1 - 5.64e6T + 4.42e13T^{2} \)
97 \( 1 + 1.20e7iT - 8.07e13T^{2} \)
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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

−21.95665282550762137301306511261, −20.87278944027108920664894374431, −18.88262392339309439870871678229, −18.43525749638968060643476734999, −15.69802679104999247921233670354, −13.48164133238546539334471703128, −11.84743279878651126854712722095, −10.20736816096031470899008223857, −6.97932163741599644036070994851, −2.36762099323966967704701167142, 5.26655570999050915116024504237, 7.969542332836451414833537033284, 10.34872906309519956109009317142, 13.09744690885587990193443742031, 15.33709212966253016823854276426, 16.22102514609993430117154380197, 17.63422327080422375084829883139, 20.26790337984979037705251944296, 21.25219848324213782039932366887, 23.37931203420390293244727689008

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