Properties

Degree $2$
Conductor $25$
Sign $-0.894 + 0.447i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.2i·2-s + 5.52i·3-s − 73.3·4-s − 56.6·6-s + 68.9i·7-s − 423. i·8-s + 212.·9-s − 486.·11-s − 404. i·12-s + 428. i·13-s − 707.·14-s + 2.00e3·16-s + 1.80e3i·17-s + 2.18e3i·18-s + 1.04e3·19-s + ⋯
L(s)  = 1  + 1.81i·2-s + 0.354i·3-s − 2.29·4-s − 0.642·6-s + 0.531i·7-s − 2.34i·8-s + 0.874·9-s − 1.21·11-s − 0.811i·12-s + 0.703i·13-s − 0.964·14-s + 1.95·16-s + 1.51i·17-s + 1.58i·18-s + 0.665·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

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

Particular Values

\(L(3)\) \(\approx\) \(0.258540 - 1.09519i\)
\(L(\frac12)\) \(\approx\) \(0.258540 - 1.09519i\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.2iT - 32T^{2} \)
3 \( 1 - 5.52iT - 243T^{2} \)
7 \( 1 - 68.9iT - 1.68e4T^{2} \)
11 \( 1 + 486.T + 1.61e5T^{2} \)
13 \( 1 - 428. iT - 3.71e5T^{2} \)
17 \( 1 - 1.80e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.04e3T + 2.47e6T^{2} \)
23 \( 1 + 686. iT - 6.43e6T^{2} \)
29 \( 1 - 1.33e3T + 2.05e7T^{2} \)
31 \( 1 - 7.99e3T + 2.86e7T^{2} \)
37 \( 1 + 1.97e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.07e4T + 1.15e8T^{2} \)
43 \( 1 + 1.50e4iT - 1.47e8T^{2} \)
47 \( 1 - 895. iT - 2.29e8T^{2} \)
53 \( 1 - 1.93e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.11e4T + 7.14e8T^{2} \)
61 \( 1 + 2.77e4T + 8.44e8T^{2} \)
67 \( 1 + 7.71e3iT - 1.35e9T^{2} \)
71 \( 1 + 5.14e4T + 1.80e9T^{2} \)
73 \( 1 - 4.37e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.22e3T + 3.07e9T^{2} \)
83 \( 1 + 5.29e4iT - 3.93e9T^{2} \)
89 \( 1 + 4.46e4T + 5.58e9T^{2} \)
97 \( 1 + 1.48e5iT - 8.58e9T^{2} \)
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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

−16.90571420575665478102109121817, −15.75470738141161281502677871300, −15.29199420958373474019884720908, −13.89973710065286387148813923578, −12.66477025061461029138078508074, −10.20211745679126918043741198789, −8.764450992946443129523729325976, −7.48600928504577769048600575301, −5.96381000751118474725766750654, −4.50796753060133859929917454680, 0.830399145543979191803325939975, 2.84117052438160518418823621882, 4.76564772446388301793929903484, 7.72132452802252651792505504663, 9.672876579993069266008313221120, 10.57556128662513838306914408334, 11.90523044566214385587897925862, 13.07250705823983688589885200640, 13.76460678795824617108157744179, 15.83915321866297565071725416832

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