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Properties

Label 2-5-5.2-c34-0-8
Degree $2$
Conductor $5$
Sign $-0.852 - 0.523i$
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  + (1.17e5 + 1.17e5i)2-s + (−6.15e7 + 6.15e7i)3-s + 1.03e10i·4-s + (3.42e11 + 6.81e11i)5-s − 1.44e13·6-s + (2.43e14 + 2.43e14i)7-s + (7.99e14 − 7.99e14i)8-s + 9.09e15i·9-s + (−3.97e16 + 1.20e17i)10-s + 5.26e17·11-s + (−6.38e17 − 6.38e17i)12-s + (8.85e18 − 8.85e18i)13-s + 5.71e19i·14-s + (−6.30e19 − 2.08e19i)15-s + 3.65e20·16-s + (−2.19e20 − 2.19e20i)17-s + ⋯
L(s)  = 1  + (0.895 + 0.895i)2-s + (−0.476 + 0.476i)3-s + 0.603i·4-s + (0.449 + 0.893i)5-s − 0.853·6-s + (1.04 + 1.04i)7-s + (0.354 − 0.354i)8-s + 0.545i·9-s + (−0.397 + 1.20i)10-s + 1.04·11-s + (−0.287 − 0.287i)12-s + (1.02 − 1.02i)13-s + 1.87i·14-s + (−0.640 − 0.211i)15-s + 1.23·16-s + (−0.265 − 0.265i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.852 - 0.523i$
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: \((2,\ 5,\ (\ :17),\ -0.852 - 0.523i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(3.908201472\)
\(L(\frac12)\) \(\approx\) \(3.908201472\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.42e11 - 6.81e11i)T \)
good2 \( 1 + (-1.17e5 - 1.17e5i)T + 1.71e10iT^{2} \)
3 \( 1 + (6.15e7 - 6.15e7i)T - 1.66e16iT^{2} \)
7 \( 1 + (-2.43e14 - 2.43e14i)T + 5.41e28iT^{2} \)
11 \( 1 - 5.26e17T + 2.55e35T^{2} \)
13 \( 1 + (-8.85e18 + 8.85e18i)T - 7.48e37iT^{2} \)
17 \( 1 + (2.19e20 + 2.19e20i)T + 6.84e41iT^{2} \)
19 \( 1 - 9.45e21iT - 3.00e43T^{2} \)
23 \( 1 + (6.63e22 - 6.63e22i)T - 1.98e46iT^{2} \)
29 \( 1 + 1.36e25iT - 5.26e49T^{2} \)
31 \( 1 - 2.68e24T + 5.08e50T^{2} \)
37 \( 1 + (2.28e26 + 2.28e26i)T + 2.08e53iT^{2} \)
41 \( 1 + 2.81e27T + 6.83e54T^{2} \)
43 \( 1 + (-2.01e27 + 2.01e27i)T - 3.45e55iT^{2} \)
47 \( 1 + (4.71e27 + 4.71e27i)T + 7.10e56iT^{2} \)
53 \( 1 + (-3.95e28 + 3.95e28i)T - 4.22e58iT^{2} \)
59 \( 1 + 2.14e29iT - 1.61e60T^{2} \)
61 \( 1 - 1.24e30T + 5.02e60T^{2} \)
67 \( 1 + (-4.25e30 - 4.25e30i)T + 1.22e62iT^{2} \)
71 \( 1 + 2.53e31T + 8.76e62T^{2} \)
73 \( 1 + (-1.19e31 + 1.19e31i)T - 2.25e63iT^{2} \)
79 \( 1 - 8.04e31iT - 3.30e64T^{2} \)
83 \( 1 + (-2.20e32 + 2.20e32i)T - 1.77e65iT^{2} \)
89 \( 1 + 1.91e33iT - 1.90e66T^{2} \)
97 \( 1 + (-4.01e33 - 4.01e33i)T + 3.55e67iT^{2} \)
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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

−15.77417790624111771595025943991, −14.78539006886310762442555377754, −13.71900052035521204603690210362, −11.64598834422414360631831781443, −10.23063382719038868583740192910, −7.977748914823655149257763682475, −6.09045594297612352369952673469, −5.45138456444416338106994957273, −3.86356003497388644639867282700, −1.79544964378694949826099942336, 1.04144351106757433288524934703, 1.62606721261894624317526208833, 3.89610053577095110952918283512, 4.86184242214584854059570670811, 6.68630120459560115820642256005, 8.805940362658045829067672299423, 11.01158795306412578302104528784, 11.93847736308267012208875966343, 13.27415745148086228521869096678, 14.23019731026032873455425820955

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