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Properties

Label 2-5-5.2-c34-0-5
Degree $2$
Conductor $5$
Sign $0.550 - 0.834i$
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  + (6.79e4 + 6.79e4i)2-s + (−7.73e7 + 7.73e7i)3-s − 7.94e9i·4-s + (−6.92e11 + 3.20e11i)5-s − 1.05e13·6-s + (−2.59e14 − 2.59e14i)7-s + (1.70e15 − 1.70e15i)8-s + 4.70e15i·9-s + (−6.88e16 − 2.52e16i)10-s − 8.17e16·11-s + (6.14e17 + 6.14e17i)12-s + (1.39e18 − 1.39e18i)13-s − 3.52e19i·14-s + (2.87e19 − 7.83e19i)15-s + 9.55e19·16-s + (3.22e20 + 3.22e20i)17-s + ⋯
L(s)  = 1  + (0.518 + 0.518i)2-s + (−0.599 + 0.599i)3-s − 0.462i·4-s + (−0.907 + 0.420i)5-s − 0.621·6-s + (−1.11 − 1.11i)7-s + (0.758 − 0.758i)8-s + 0.281i·9-s + (−0.688 − 0.252i)10-s − 0.161·11-s + (0.277 + 0.277i)12-s + (0.161 − 0.161i)13-s − 1.15i·14-s + (0.291 − 0.795i)15-s + 0.323·16-s + (0.389 + 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.550 - 0.834i$
Analytic conductor: \(36.6128\)
Root analytic conductor: \(6.05085\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :17),\ 0.550 - 0.834i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(1.307455517\)
\(L(\frac12)\) \(\approx\) \(1.307455517\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (6.92e11 - 3.20e11i)T \)
good2 \( 1 + (-6.79e4 - 6.79e4i)T + 1.71e10iT^{2} \)
3 \( 1 + (7.73e7 - 7.73e7i)T - 1.66e16iT^{2} \)
7 \( 1 + (2.59e14 + 2.59e14i)T + 5.41e28iT^{2} \)
11 \( 1 + 8.17e16T + 2.55e35T^{2} \)
13 \( 1 + (-1.39e18 + 1.39e18i)T - 7.48e37iT^{2} \)
17 \( 1 + (-3.22e20 - 3.22e20i)T + 6.84e41iT^{2} \)
19 \( 1 - 2.60e21iT - 3.00e43T^{2} \)
23 \( 1 + (-5.14e22 + 5.14e22i)T - 1.98e46iT^{2} \)
29 \( 1 - 1.01e25iT - 5.26e49T^{2} \)
31 \( 1 + 9.26e24T + 5.08e50T^{2} \)
37 \( 1 + (-4.80e26 - 4.80e26i)T + 2.08e53iT^{2} \)
41 \( 1 - 2.99e27T + 6.83e54T^{2} \)
43 \( 1 + (3.50e27 - 3.50e27i)T - 3.45e55iT^{2} \)
47 \( 1 + (-3.01e28 - 3.01e28i)T + 7.10e56iT^{2} \)
53 \( 1 + (-1.96e29 + 1.96e29i)T - 4.22e58iT^{2} \)
59 \( 1 + 1.74e30iT - 1.61e60T^{2} \)
61 \( 1 - 2.76e29T + 5.02e60T^{2} \)
67 \( 1 + (-6.56e30 - 6.56e30i)T + 1.22e62iT^{2} \)
71 \( 1 + 4.26e31T + 8.76e62T^{2} \)
73 \( 1 + (-4.46e31 + 4.46e31i)T - 2.25e63iT^{2} \)
79 \( 1 + 2.33e32iT - 3.30e64T^{2} \)
83 \( 1 + (1.29e32 - 1.29e32i)T - 1.77e65iT^{2} \)
89 \( 1 + 1.79e33iT - 1.90e66T^{2} \)
97 \( 1 + (-8.38e33 - 8.38e33i)T + 3.55e67iT^{2} \)
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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

−15.95441314093536715111795695070, −14.53905590822163769354731625583, −13.00568942236580635258333733125, −10.89491725106294218258312611383, −10.09346727523513565569599659725, −7.45585920970312263990437220969, −6.20821771928410604889674654532, −4.64112790988592366666134492828, −3.52907843049405663653768931852, −0.72549604166868078611249135292, 0.57728684282144823061450745355, 2.58373458573579441344864474861, 3.88013769832842292418341715122, 5.65535186387574686182734241964, 7.32855251868612574074963129366, 9.019124413820991836641759947473, 11.50360756627660531051357205091, 12.26611567160749834637110032618, 13.11268977860637435829335803916, 15.48274282346960419045037116278

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