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Properties

Label 2-5-5.2-c34-0-10
Degree $2$
Conductor $5$
Sign $0.832 - 0.553i$
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  + (1.36e5 + 1.36e5i)2-s + (1.30e8 − 1.30e8i)3-s + 2.03e10i·4-s + (−7.62e11 + 2.54e10i)5-s + 3.56e13·6-s + (1.04e14 + 1.04e14i)7-s + (−4.32e14 + 4.32e14i)8-s − 1.72e16i·9-s + (−1.07e17 − 1.00e17i)10-s + 6.99e17·11-s + (2.64e18 + 2.64e18i)12-s + (−2.18e18 + 2.18e18i)13-s + 2.85e19i·14-s + (−9.59e19 + 1.02e20i)15-s + 2.30e20·16-s + (8.87e20 + 8.87e20i)17-s + ⋯
L(s)  = 1  + (1.04 + 1.04i)2-s + (1.00 − 1.00i)3-s + 1.18i·4-s + (−0.999 + 0.0333i)5-s + 2.10·6-s + (0.447 + 0.447i)7-s + (−0.192 + 0.192i)8-s − 1.03i·9-s + (−1.07 − 1.00i)10-s + 1.38·11-s + (1.19 + 1.19i)12-s + (−0.253 + 0.253i)13-s + 0.935i·14-s + (−0.974 + 1.04i)15-s + 0.782·16-s + (1.07 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.832 - 0.553i$
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: \((2,\ 5,\ (\ :17),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(5.152068329\)
\(L(\frac12)\) \(\approx\) \(5.152068329\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (7.62e11 - 2.54e10i)T \)
good2 \( 1 + (-1.36e5 - 1.36e5i)T + 1.71e10iT^{2} \)
3 \( 1 + (-1.30e8 + 1.30e8i)T - 1.66e16iT^{2} \)
7 \( 1 + (-1.04e14 - 1.04e14i)T + 5.41e28iT^{2} \)
11 \( 1 - 6.99e17T + 2.55e35T^{2} \)
13 \( 1 + (2.18e18 - 2.18e18i)T - 7.48e37iT^{2} \)
17 \( 1 + (-8.87e20 - 8.87e20i)T + 6.84e41iT^{2} \)
19 \( 1 + 7.09e21iT - 3.00e43T^{2} \)
23 \( 1 + (-9.94e22 + 9.94e22i)T - 1.98e46iT^{2} \)
29 \( 1 + 5.08e24iT - 5.26e49T^{2} \)
31 \( 1 + 1.13e25T + 5.08e50T^{2} \)
37 \( 1 + (-4.09e26 - 4.09e26i)T + 2.08e53iT^{2} \)
41 \( 1 + 1.39e26T + 6.83e54T^{2} \)
43 \( 1 + (3.34e27 - 3.34e27i)T - 3.45e55iT^{2} \)
47 \( 1 + (2.50e28 + 2.50e28i)T + 7.10e56iT^{2} \)
53 \( 1 + (2.37e29 - 2.37e29i)T - 4.22e58iT^{2} \)
59 \( 1 - 1.24e30iT - 1.61e60T^{2} \)
61 \( 1 + 2.53e30T + 5.02e60T^{2} \)
67 \( 1 + (-2.31e30 - 2.31e30i)T + 1.22e62iT^{2} \)
71 \( 1 + 1.82e30T + 8.76e62T^{2} \)
73 \( 1 + (6.38e31 - 6.38e31i)T - 2.25e63iT^{2} \)
79 \( 1 + 2.50e32iT - 3.30e64T^{2} \)
83 \( 1 + (-3.61e32 + 3.61e32i)T - 1.77e65iT^{2} \)
89 \( 1 + 1.24e33iT - 1.90e66T^{2} \)
97 \( 1 + (2.98e33 + 2.98e33i)T + 3.55e67iT^{2} \)
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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

−15.00319024595412977390636540628, −14.53210944250555822779532931477, −13.07469866972435012422357329023, −11.87892887323713988105162541683, −8.634246440576161249250171943127, −7.54557748832049967700886258204, −6.48953205727320322068361386367, −4.51325891287057848156239392983, −3.16641479563303693014813045902, −1.29126486915340219029691361108, 1.30359011030098552619110503734, 3.21024001217837623474167119342, 3.78291358538130251256788073163, 4.86762416382560581797495121894, 7.81553002960589734032001227108, 9.548838603351864287541868710519, 11.07720628850901599969114920654, 12.26003052397431268102568577271, 14.21918566750947242952851539197, 14.73066769364768956865849498771

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