Properties

Label 2-5-5.3-c34-0-11
Degree $2$
Conductor $5$
Sign $-0.00796 + 0.999i$
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  + (−1.69e5 + 1.69e5i)2-s + (−1.98e5 − 1.98e5i)3-s − 4.02e10i·4-s + (−3.95e11 − 6.52e11i)5-s + 6.73e10·6-s + (1.10e14 − 1.10e14i)7-s + (3.91e15 + 3.91e15i)8-s − 1.66e16i·9-s + (1.77e17 + 4.34e16i)10-s + 8.70e17·11-s + (−8.00e15 + 8.00e15i)12-s + (−7.62e18 − 7.62e18i)13-s + 3.76e19i·14-s + (−5.09e16 + 2.08e17i)15-s − 6.35e20·16-s + (8.05e19 − 8.05e19i)17-s + ⋯
L(s)  = 1  + (−1.29 + 1.29i)2-s + (−0.00153 − 0.00153i)3-s − 2.34i·4-s + (−0.518 − 0.854i)5-s + 0.00397·6-s + (0.477 − 0.477i)7-s + (1.73 + 1.73i)8-s − 0.999i·9-s + (1.77 + 0.434i)10-s + 1.72·11-s + (−0.00360 + 0.00360i)12-s + (−0.881 − 0.881i)13-s + 1.23i·14-s + (−0.000516 + 0.00211i)15-s − 2.15·16-s + (0.0974 − 0.0974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00796 + 0.999i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.00796 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.00796 + 0.999i$
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: \((2,\ 5,\ (\ :17),\ -0.00796 + 0.999i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.7483959418\)
\(L(\frac12)\) \(\approx\) \(0.7483959418\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.95e11 + 6.52e11i)T \)
good2 \( 1 + (1.69e5 - 1.69e5i)T - 1.71e10iT^{2} \)
3 \( 1 + (1.98e5 + 1.98e5i)T + 1.66e16iT^{2} \)
7 \( 1 + (-1.10e14 + 1.10e14i)T - 5.41e28iT^{2} \)
11 \( 1 - 8.70e17T + 2.55e35T^{2} \)
13 \( 1 + (7.62e18 + 7.62e18i)T + 7.48e37iT^{2} \)
17 \( 1 + (-8.05e19 + 8.05e19i)T - 6.84e41iT^{2} \)
19 \( 1 + 7.48e20iT - 3.00e43T^{2} \)
23 \( 1 + (7.83e22 + 7.83e22i)T + 1.98e46iT^{2} \)
29 \( 1 + 9.02e24iT - 5.26e49T^{2} \)
31 \( 1 - 3.35e25T + 5.08e50T^{2} \)
37 \( 1 + (1.09e26 - 1.09e26i)T - 2.08e53iT^{2} \)
41 \( 1 - 3.70e27T + 6.83e54T^{2} \)
43 \( 1 + (2.65e27 + 2.65e27i)T + 3.45e55iT^{2} \)
47 \( 1 + (1.63e27 - 1.63e27i)T - 7.10e56iT^{2} \)
53 \( 1 + (-1.90e29 - 1.90e29i)T + 4.22e58iT^{2} \)
59 \( 1 + 8.59e29iT - 1.61e60T^{2} \)
61 \( 1 - 7.88e29T + 5.02e60T^{2} \)
67 \( 1 + (-6.99e30 + 6.99e30i)T - 1.22e62iT^{2} \)
71 \( 1 + 4.47e31T + 8.76e62T^{2} \)
73 \( 1 + (2.85e31 + 2.85e31i)T + 2.25e63iT^{2} \)
79 \( 1 - 5.01e30iT - 3.30e64T^{2} \)
83 \( 1 + (4.76e32 + 4.76e32i)T + 1.77e65iT^{2} \)
89 \( 1 + 2.19e33iT - 1.90e66T^{2} \)
97 \( 1 + (6.21e33 - 6.21e33i)T - 3.55e67iT^{2} \)
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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

−15.51255834452024702837441701592, −14.42541092437525849550004967122, −11.91459689071128046442224556774, −9.830130041065017609109693266132, −8.722359881529263291840563984709, −7.52531153406731116435109740112, −6.15573283235106493555334685290, −4.38786159191232065463373312313, −1.13310942472097959525434480920, −0.42426629083747930906171719622, 1.45746739694395356310263508088, 2.53050858844511970505051728080, 4.06635040904545452260135291232, 7.14225294412300109461476970025, 8.509224387160687483205423647268, 9.908743793150572909101597771761, 11.32248547473880106054690952623, 11.99811688234494878108867988173, 14.35550243985068139343488771488, 16.56110131058668798979700897955

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