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Properties

Label 2-5-5.3-c16-0-4
Degree $2$
Conductor $5$
Sign $0.870 - 0.492i$
Analytic cond. $8.11622$
Root an. cond. $2.84889$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−330. + 330. i)2-s + (4.93e3 + 4.93e3i)3-s − 1.52e5i·4-s + (1.88e5 − 3.42e5i)5-s − 3.26e6·6-s + (5.35e6 − 5.35e6i)7-s + (2.87e7 + 2.87e7i)8-s + 5.74e6i·9-s + (5.09e7 + 1.75e8i)10-s − 2.19e7·11-s + (7.54e8 − 7.54e8i)12-s + (−5.84e8 − 5.84e8i)13-s + 3.53e9i·14-s + (2.61e9 − 7.62e8i)15-s − 9.00e9·16-s + (3.70e9 − 3.70e9i)17-s + ⋯
L(s)  = 1  + (−1.29 + 1.29i)2-s + (0.752 + 0.752i)3-s − 2.32i·4-s + (0.481 − 0.876i)5-s − 1.94·6-s + (0.928 − 0.928i)7-s + (1.71 + 1.71i)8-s + 0.133i·9-s + (0.509 + 1.75i)10-s − 0.102·11-s + (1.75 − 1.75i)12-s + (−0.715 − 0.715i)13-s + 2.39i·14-s + (1.02 − 0.297i)15-s − 2.09·16-s + (0.530 − 0.530i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.870 - 0.492i$
Analytic conductor: \(8.11622\)
Root analytic conductor: \(2.84889\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :8),\ 0.870 - 0.492i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(1.17844 + 0.310400i\)
\(L(\frac12)\) \(\approx\) \(1.17844 + 0.310400i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.88e5 + 3.42e5i)T \)
good2 \( 1 + (330. - 330. i)T - 6.55e4iT^{2} \)
3 \( 1 + (-4.93e3 - 4.93e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (-5.35e6 + 5.35e6i)T - 3.32e13iT^{2} \)
11 \( 1 + 2.19e7T + 4.59e16T^{2} \)
13 \( 1 + (5.84e8 + 5.84e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (-3.70e9 + 3.70e9i)T - 4.86e19iT^{2} \)
19 \( 1 + 2.81e9iT - 2.88e20T^{2} \)
23 \( 1 + (-6.91e10 - 6.91e10i)T + 6.13e21iT^{2} \)
29 \( 1 - 1.74e11iT - 2.50e23T^{2} \)
31 \( 1 + 1.63e11T + 7.27e23T^{2} \)
37 \( 1 + (2.94e12 - 2.94e12i)T - 1.23e25iT^{2} \)
41 \( 1 - 5.12e12T + 6.37e25T^{2} \)
43 \( 1 + (-6.12e12 - 6.12e12i)T + 1.36e26iT^{2} \)
47 \( 1 + (-2.97e13 + 2.97e13i)T - 5.66e26iT^{2} \)
53 \( 1 + (4.13e13 + 4.13e13i)T + 3.87e27iT^{2} \)
59 \( 1 - 1.91e14iT - 2.15e28T^{2} \)
61 \( 1 + 1.91e14T + 3.67e28T^{2} \)
67 \( 1 + (-1.50e14 + 1.50e14i)T - 1.64e29iT^{2} \)
71 \( 1 + 4.21e14T + 4.16e29T^{2} \)
73 \( 1 + (-2.37e14 - 2.37e14i)T + 6.50e29iT^{2} \)
79 \( 1 - 9.75e13iT - 2.30e30T^{2} \)
83 \( 1 + (-1.15e15 - 1.15e15i)T + 5.07e30iT^{2} \)
89 \( 1 - 2.17e15iT - 1.54e31T^{2} \)
97 \( 1 + (-9.14e15 + 9.14e15i)T - 6.14e31iT^{2} \)
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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

−19.84532479488798177456225039671, −17.75876063569975017631915864748, −16.81307833940810043408679462207, −15.34775702304187631360877953329, −14.10135557331805276156038128952, −10.19258676189460113891337879659, −9.001097506341925256320675815553, −7.63316192277026554291783392218, −5.06465533628580331732801507528, −0.960897831969735379012764956905, 1.76602115989685963906017089087, 2.63260052953291424265924496484, 7.61432766481295639410808459623, 9.030287885683539324085619998357, 10.82477604540128546495243694579, 12.39777822142633571050884621238, 14.38197605205786103262798523201, 17.38351501978460088717886880902, 18.65795400887787679076924286111, 19.14476679909297518581464091904

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