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Properties

Label 2-5e2-5.4-c9-0-11
Degree $2$
Conductor $25$
Sign $0.447 - 0.894i$
Analytic cond. $12.8758$
Root an. cond. $3.58829$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 36.7i·2-s − 193. i·3-s − 839.·4-s − 7.11e3·6-s + 7.64e3i·7-s + 1.20e4i·8-s − 1.77e4·9-s − 4.83e4·11-s + 1.62e5i·12-s − 1.00e5i·13-s + 2.81e5·14-s + 1.29e4·16-s − 2.01e5i·17-s + 6.53e5i·18-s + 5.80e4·19-s + ⋯
L(s)  = 1  − 1.62i·2-s − 1.37i·3-s − 1.63·4-s − 2.24·6-s + 1.20i·7-s + 1.03i·8-s − 0.902·9-s − 0.995·11-s + 2.26i·12-s − 0.975i·13-s + 1.95·14-s + 0.0494·16-s − 0.586i·17-s + 1.46i·18-s + 0.102·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(12.8758\)
Root analytic conductor: \(3.58829\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :9/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.619510 + 0.382878i\)
\(L(\frac12)\) \(\approx\) \(0.619510 + 0.382878i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + 36.7iT - 512T^{2} \)
3 \( 1 + 193. iT - 1.96e4T^{2} \)
7 \( 1 - 7.64e3iT - 4.03e7T^{2} \)
11 \( 1 + 4.83e4T + 2.35e9T^{2} \)
13 \( 1 + 1.00e5iT - 1.06e10T^{2} \)
17 \( 1 + 2.01e5iT - 1.18e11T^{2} \)
19 \( 1 - 5.80e4T + 3.22e11T^{2} \)
23 \( 1 - 1.14e6iT - 1.80e12T^{2} \)
29 \( 1 + 1.56e6T + 1.45e13T^{2} \)
31 \( 1 + 4.10e6T + 2.64e13T^{2} \)
37 \( 1 + 1.70e7iT - 1.29e14T^{2} \)
41 \( 1 + 8.52e6T + 3.27e14T^{2} \)
43 \( 1 + 2.56e7iT - 5.02e14T^{2} \)
47 \( 1 - 4.58e7iT - 1.11e15T^{2} \)
53 \( 1 - 5.56e7iT - 3.29e15T^{2} \)
59 \( 1 + 2.15e7T + 8.66e15T^{2} \)
61 \( 1 + 1.15e8T + 1.16e16T^{2} \)
67 \( 1 + 7.69e7iT - 2.72e16T^{2} \)
71 \( 1 - 1.95e8T + 4.58e16T^{2} \)
73 \( 1 + 3.40e8iT - 5.88e16T^{2} \)
79 \( 1 - 5.92e8T + 1.19e17T^{2} \)
83 \( 1 + 7.88e8iT - 1.86e17T^{2} \)
89 \( 1 + 8.40e8T + 3.50e17T^{2} \)
97 \( 1 - 2.35e8iT - 7.60e17T^{2} \)
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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

−13.66869794786887878151514669117, −12.72485826175055048949770228990, −12.11389574970615256089201699061, −10.83476683827091860865742793696, −9.206890255828944667739585470505, −7.67610620434959451470604276536, −5.52661825788034724275168706401, −2.91880463612917603115666935794, −1.87753899053988584848429909115, −0.30271225169397437896246474325, 4.03381310156870537761262334136, 5.09275597315319984066603684366, 6.81577969044807745803920107415, 8.248098838174145416956785821016, 9.701680975958107520650888831269, 10.85743402260094624593961821514, 13.39129005535200037789200891597, 14.52006370341208821225364396709, 15.43254390590785474421228679896, 16.52284078079503277047083276309

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