Properties

Label 2-5-5.3-c34-0-13
Degree $2$
Conductor $5$
Sign $-0.432 + 0.901i$
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  + (8.62e3 − 8.62e3i)2-s + (9.89e7 + 9.89e7i)3-s + 1.70e10i·4-s + (−8.06e10 − 7.58e11i)5-s + 1.70e12·6-s + (1.10e14 − 1.10e14i)7-s + (2.95e14 + 2.95e14i)8-s + 2.88e15i·9-s + (−7.24e15 − 5.84e15i)10-s − 6.88e17·11-s + (−1.68e18 + 1.68e18i)12-s + (−6.48e18 − 6.48e18i)13-s − 1.91e18i·14-s + (6.70e19 − 8.30e19i)15-s − 2.87e20·16-s + (−2.93e20 + 2.93e20i)17-s + ⋯
L(s)  = 1  + (0.0658 − 0.0658i)2-s + (0.765 + 0.765i)3-s + 0.991i·4-s + (−0.105 − 0.994i)5-s + 0.100·6-s + (0.476 − 0.476i)7-s + (0.131 + 0.131i)8-s + 0.173i·9-s + (−0.0724 − 0.0584i)10-s − 1.36·11-s + (−0.759 + 0.759i)12-s + (−0.750 − 0.750i)13-s − 0.0627i·14-s + (0.680 − 0.842i)15-s − 0.974·16-s + (−0.354 + 0.354i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.432 + 0.901i$
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: \((2,\ 5,\ (\ :17),\ -0.432 + 0.901i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.8282603649\)
\(L(\frac12)\) \(\approx\) \(0.8282603649\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (8.06e10 + 7.58e11i)T \)
good2 \( 1 + (-8.62e3 + 8.62e3i)T - 1.71e10iT^{2} \)
3 \( 1 + (-9.89e7 - 9.89e7i)T + 1.66e16iT^{2} \)
7 \( 1 + (-1.10e14 + 1.10e14i)T - 5.41e28iT^{2} \)
11 \( 1 + 6.88e17T + 2.55e35T^{2} \)
13 \( 1 + (6.48e18 + 6.48e18i)T + 7.48e37iT^{2} \)
17 \( 1 + (2.93e20 - 2.93e20i)T - 6.84e41iT^{2} \)
19 \( 1 - 1.21e21iT - 3.00e43T^{2} \)
23 \( 1 + (1.01e23 + 1.01e23i)T + 1.98e46iT^{2} \)
29 \( 1 - 1.11e23iT - 5.26e49T^{2} \)
31 \( 1 - 5.21e24T + 5.08e50T^{2} \)
37 \( 1 + (-3.43e26 + 3.43e26i)T - 2.08e53iT^{2} \)
41 \( 1 + 3.54e27T + 6.83e54T^{2} \)
43 \( 1 + (7.17e27 + 7.17e27i)T + 3.45e55iT^{2} \)
47 \( 1 + (-1.36e28 + 1.36e28i)T - 7.10e56iT^{2} \)
53 \( 1 + (2.76e28 + 2.76e28i)T + 4.22e58iT^{2} \)
59 \( 1 - 1.87e30iT - 1.61e60T^{2} \)
61 \( 1 - 2.15e30T + 5.02e60T^{2} \)
67 \( 1 + (6.36e30 - 6.36e30i)T - 1.22e62iT^{2} \)
71 \( 1 - 5.86e31T + 8.76e62T^{2} \)
73 \( 1 + (5.09e31 + 5.09e31i)T + 2.25e63iT^{2} \)
79 \( 1 + 3.07e32iT - 3.30e64T^{2} \)
83 \( 1 + (-2.32e32 - 2.32e32i)T + 1.77e65iT^{2} \)
89 \( 1 - 2.13e33iT - 1.90e66T^{2} \)
97 \( 1 + (-2.18e33 + 2.18e33i)T - 3.55e67iT^{2} \)
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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

−15.25767205189090815684292199248, −13.47213553888872695662243647963, −12.26064678972042997996168653338, −10.25575571684940985391088256358, −8.609671161907087791164460563487, −7.78003455686554666620082538127, −4.93085144524915651035295224932, −3.84403109956904798396832792088, −2.42877056733720794014044835067, −0.19248735591042260209229275586, 1.84060162403662717268775705359, 2.64932531366238405213734899123, 5.06246117796117957310910835463, 6.75496803162955576226733923635, 8.009626143831810923153972130096, 9.897163796856779111816416124082, 11.37530655489444828066681076744, 13.46966954858052030709336376292, 14.46337257509999394074295231831, 15.53523906526692155797739025841

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