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Properties

Label 2-5-5.3-c34-0-0
Degree $2$
Conductor $5$
Sign $-0.991 + 0.128i$
Analytic cond. $36.6128$
Root an. cond. $6.05085$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.32e4 + 8.32e4i)2-s + (−2.65e7 − 2.65e7i)3-s + 3.31e9i·4-s + (5.91e11 − 4.81e11i)5-s + 4.41e12·6-s + (−1.23e14 + 1.23e14i)7-s + (−1.70e15 − 1.70e15i)8-s − 1.52e16i·9-s + (−9.21e15 + 8.93e16i)10-s + 1.69e17·11-s + (8.79e16 − 8.79e16i)12-s + (3.25e18 + 3.25e18i)13-s − 2.05e19i·14-s + (−2.84e19 − 2.93e18i)15-s + 2.27e20·16-s + (−2.13e20 + 2.13e20i)17-s + ⋯
L(s)  = 1  + (−0.635 + 0.635i)2-s + (−0.205 − 0.205i)3-s + 0.193i·4-s + (0.775 − 0.630i)5-s + 0.260·6-s + (−0.530 + 0.530i)7-s + (−0.757 − 0.757i)8-s − 0.915i·9-s + (−0.0921 + 0.893i)10-s + 0.334·11-s + (0.0396 − 0.0396i)12-s + (0.376 + 0.376i)13-s − 0.674i·14-s + (−0.288 − 0.0297i)15-s + 0.769·16-s + (−0.258 + 0.258i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.991 + 0.128i$
Analytic conductor: \(36.6128\)
Root analytic conductor: \(6.05085\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :17),\ -0.991 + 0.128i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.3416664784\)
\(L(\frac12)\) \(\approx\) \(0.3416664784\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-5.91e11 + 4.81e11i)T \)
good2 \( 1 + (8.32e4 - 8.32e4i)T - 1.71e10iT^{2} \)
3 \( 1 + (2.65e7 + 2.65e7i)T + 1.66e16iT^{2} \)
7 \( 1 + (1.23e14 - 1.23e14i)T - 5.41e28iT^{2} \)
11 \( 1 - 1.69e17T + 2.55e35T^{2} \)
13 \( 1 + (-3.25e18 - 3.25e18i)T + 7.48e37iT^{2} \)
17 \( 1 + (2.13e20 - 2.13e20i)T - 6.84e41iT^{2} \)
19 \( 1 - 3.34e21iT - 3.00e43T^{2} \)
23 \( 1 + (-1.84e23 - 1.84e23i)T + 1.98e46iT^{2} \)
29 \( 1 + 5.36e24iT - 5.26e49T^{2} \)
31 \( 1 + 3.82e25T + 5.08e50T^{2} \)
37 \( 1 + (3.49e26 - 3.49e26i)T - 2.08e53iT^{2} \)
41 \( 1 + 2.58e27T + 6.83e54T^{2} \)
43 \( 1 + (5.57e27 + 5.57e27i)T + 3.45e55iT^{2} \)
47 \( 1 + (4.46e27 - 4.46e27i)T - 7.10e56iT^{2} \)
53 \( 1 + (8.23e27 + 8.23e27i)T + 4.22e58iT^{2} \)
59 \( 1 - 5.95e28iT - 1.61e60T^{2} \)
61 \( 1 - 3.53e28T + 5.02e60T^{2} \)
67 \( 1 + (4.37e30 - 4.37e30i)T - 1.22e62iT^{2} \)
71 \( 1 + 3.51e31T + 8.76e62T^{2} \)
73 \( 1 + (2.62e31 + 2.62e31i)T + 2.25e63iT^{2} \)
79 \( 1 - 1.83e32iT - 3.30e64T^{2} \)
83 \( 1 + (-3.82e32 - 3.82e32i)T + 1.77e65iT^{2} \)
89 \( 1 + 1.89e33iT - 1.90e66T^{2} \)
97 \( 1 + (3.99e33 - 3.99e33i)T - 3.55e67iT^{2} \)
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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

−16.66380297109121530794616630418, −15.26233314559490902545276874983, −13.16797717641936176281623827233, −12.01714590503241332691969575181, −9.534063117975705682712339079157, −8.757249565982185582617503795226, −6.85172310728219332177705669732, −5.76504884495583501422174337036, −3.48483806076239141889020197705, −1.41398618717646055413427939468, 0.12882983638317997348031796751, 1.65765190699612058799923926395, 2.98377109949420042654315874252, 5.24862977776325174253319321345, 6.79749301898277392414862023169, 9.017363171282726708920176292002, 10.36206365158217282403760601289, 11.01846074960046104431521950337, 13.26800098702635113292675048313, 14.70167531531790486092520792933

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