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Properties

Label 2-5-5.2-c34-0-14
Degree $2$
Conductor $5$
Sign $-0.450 + 0.892i$
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  + (3.66e4 + 3.66e4i)2-s + (1.35e8 − 1.35e8i)3-s − 1.44e10i·4-s + (6.50e11 − 3.99e11i)5-s + 9.95e12·6-s + (−1.47e14 − 1.47e14i)7-s + (1.16e15 − 1.16e15i)8-s − 2.02e16i·9-s + (3.84e16 + 9.20e15i)10-s + 7.69e17·11-s + (−1.96e18 − 1.96e18i)12-s + (5.59e18 − 5.59e18i)13-s − 1.08e19i·14-s + (3.41e19 − 1.42e20i)15-s − 1.64e20·16-s + (−4.36e19 − 4.36e19i)17-s + ⋯
L(s)  = 1  + (0.279 + 0.279i)2-s + (1.05 − 1.05i)3-s − 0.843i·4-s + (0.852 − 0.522i)5-s + 0.588·6-s + (−0.635 − 0.635i)7-s + (0.515 − 0.515i)8-s − 1.21i·9-s + (0.384 + 0.0920i)10-s + 1.52·11-s + (−0.887 − 0.887i)12-s + (0.646 − 0.646i)13-s − 0.355i·14-s + (0.346 − 1.44i)15-s − 0.555·16-s + (−0.0527 − 0.0527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.450 + 0.892i$
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: \((2,\ 5,\ (\ :17),\ -0.450 + 0.892i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(4.167083442\)
\(L(\frac12)\) \(\approx\) \(4.167083442\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-6.50e11 + 3.99e11i)T \)
good2 \( 1 + (-3.66e4 - 3.66e4i)T + 1.71e10iT^{2} \)
3 \( 1 + (-1.35e8 + 1.35e8i)T - 1.66e16iT^{2} \)
7 \( 1 + (1.47e14 + 1.47e14i)T + 5.41e28iT^{2} \)
11 \( 1 - 7.69e17T + 2.55e35T^{2} \)
13 \( 1 + (-5.59e18 + 5.59e18i)T - 7.48e37iT^{2} \)
17 \( 1 + (4.36e19 + 4.36e19i)T + 6.84e41iT^{2} \)
19 \( 1 - 9.91e21iT - 3.00e43T^{2} \)
23 \( 1 + (4.35e22 - 4.35e22i)T - 1.98e46iT^{2} \)
29 \( 1 - 1.16e25iT - 5.26e49T^{2} \)
31 \( 1 + 6.82e24T + 5.08e50T^{2} \)
37 \( 1 + (3.61e25 + 3.61e25i)T + 2.08e53iT^{2} \)
41 \( 1 + 1.34e27T + 6.83e54T^{2} \)
43 \( 1 + (-2.87e27 + 2.87e27i)T - 3.45e55iT^{2} \)
47 \( 1 + (6.80e27 + 6.80e27i)T + 7.10e56iT^{2} \)
53 \( 1 + (7.11e28 - 7.11e28i)T - 4.22e58iT^{2} \)
59 \( 1 - 1.99e30iT - 1.61e60T^{2} \)
61 \( 1 - 4.23e30T + 5.02e60T^{2} \)
67 \( 1 + (1.01e31 + 1.01e31i)T + 1.22e62iT^{2} \)
71 \( 1 - 8.91e30T + 8.76e62T^{2} \)
73 \( 1 + (2.78e31 - 2.78e31i)T - 2.25e63iT^{2} \)
79 \( 1 - 3.78e31iT - 3.30e64T^{2} \)
83 \( 1 + (5.24e32 - 5.24e32i)T - 1.77e65iT^{2} \)
89 \( 1 + 1.16e33iT - 1.90e66T^{2} \)
97 \( 1 + (2.55e33 + 2.55e33i)T + 3.55e67iT^{2} \)
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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

−14.47757466232178515066998797703, −13.79287408266869202493934832340, −12.68780408923338556811670286731, −10.08227937193802382100853298153, −8.780482269279880013476529263090, −6.96048291759899756716938765360, −5.86971573821426713614924314721, −3.67589324937944864231590940539, −1.66472152869936173216043344066, −1.07304075959914431616357344694, 2.20831228597541000310980383218, 3.20060771764331979102325102126, 4.29693630916752765037966904940, 6.55555168961117172484266434676, 8.796040026391033527992860110785, 9.537000705239615592226791424049, 11.42569332467883811801987374436, 13.34278515380206332547192793858, 14.43022019105782039811639570866, 15.87441091771226815929963693051

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