Properties

Label 2-5-5.4-c33-0-10
Degree $2$
Conductor $5$
Sign $0.627 - 0.778i$
Analytic cond. $34.4914$
Root an. cond. $5.87293$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56e5i·2-s + 6.96e7i·3-s − 1.57e10·4-s + (−2.14e11 + 2.65e11i)5-s − 1.08e13·6-s − 1.55e14i·7-s − 1.12e15i·8-s + 7.11e14·9-s + (−4.14e16 − 3.34e16i)10-s + 4.13e16·11-s − 1.09e18i·12-s + 2.38e18i·13-s + 2.42e19·14-s + (−1.84e19 − 1.49e19i)15-s + 3.93e19·16-s − 2.91e20i·17-s + ⋯
L(s)  = 1  + 1.68i·2-s + 0.933i·3-s − 1.83·4-s + (−0.627 + 0.778i)5-s − 1.57·6-s − 1.76i·7-s − 1.40i·8-s + 0.128·9-s + (−1.31 − 1.05i)10-s + 0.271·11-s − 1.71i·12-s + 0.993i·13-s + 2.97·14-s + (−0.726 − 0.586i)15-s + 0.533·16-s − 1.45i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.627 - 0.778i$
Analytic conductor: \(34.4914\)
Root analytic conductor: \(5.87293\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :33/2),\ 0.627 - 0.778i)\)

Particular Values

\(L(17)\) \(\approx\) \(0.8440923622\)
\(L(\frac12)\) \(\approx\) \(0.8440923622\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.14e11 - 2.65e11i)T \)
good2 \( 1 - 1.56e5iT - 8.58e9T^{2} \)
3 \( 1 - 6.96e7iT - 5.55e15T^{2} \)
7 \( 1 + 1.55e14iT - 7.73e27T^{2} \)
11 \( 1 - 4.13e16T + 2.32e34T^{2} \)
13 \( 1 - 2.38e18iT - 5.75e36T^{2} \)
17 \( 1 + 2.91e20iT - 4.02e40T^{2} \)
19 \( 1 - 3.61e20T + 1.58e42T^{2} \)
23 \( 1 + 3.81e22iT - 8.65e44T^{2} \)
29 \( 1 + 1.65e24T + 1.81e48T^{2} \)
31 \( 1 + 2.76e24T + 1.64e49T^{2} \)
37 \( 1 + 8.70e24iT - 5.63e51T^{2} \)
41 \( 1 - 2.69e26T + 1.66e53T^{2} \)
43 \( 1 - 1.33e26iT - 8.02e53T^{2} \)
47 \( 1 + 2.88e27iT - 1.51e55T^{2} \)
53 \( 1 - 2.92e28iT - 7.96e56T^{2} \)
59 \( 1 + 1.51e29T + 2.74e58T^{2} \)
61 \( 1 - 2.88e29T + 8.23e58T^{2} \)
67 \( 1 + 8.72e29iT - 1.82e60T^{2} \)
71 \( 1 - 3.08e30T + 1.23e61T^{2} \)
73 \( 1 + 2.41e30iT - 3.08e61T^{2} \)
79 \( 1 - 3.21e31T + 4.18e62T^{2} \)
83 \( 1 + 7.79e31iT - 2.13e63T^{2} \)
89 \( 1 + 1.41e32T + 2.13e64T^{2} \)
97 \( 1 - 7.13e32iT - 3.65e65T^{2} \)
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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

−16.12979880240380770564325193103, −14.72145314275537491238468675049, −13.84390223253056789807467839791, −10.93453396206855901609794914814, −9.450136055534139918412846644117, −7.48156452421808211893012611057, −6.80807642323918498788538557641, −4.64075123348243738719319160034, −3.84137935240166514637458185348, −0.28932661428998160676810426312, 1.18487300014509095919805226043, 2.09644892196567862424782551314, 3.63497276296200495581833618792, 5.50374370778754642935691524640, 8.121260604441988462987898006669, 9.382932015746195596324373729808, 11.40643986180148347604205165396, 12.49515995756446240034957728560, 12.92383368359655645332987764219, 15.37280275031650888822394203570

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