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Properties

Label 2-5-5.2-c12-0-4
Degree $2$
Conductor $5$
Sign $-0.998 + 0.0629i$
Analytic cond. $4.56996$
Root an. cond. $2.13774$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−61.7 − 61.7i)2-s + (877. − 877. i)3-s + 3.53e3i·4-s + (−1.37e4 − 7.36e3i)5-s − 1.08e5·6-s + (6.75e4 + 6.75e4i)7-s + (−3.43e4 + 3.43e4i)8-s − 1.00e6i·9-s + (3.96e5 + 1.30e6i)10-s − 6.26e5·11-s + (3.10e6 + 3.10e6i)12-s + (3.07e6 − 3.07e6i)13-s − 8.35e6i·14-s + (−1.85e7 + 5.63e6i)15-s + 1.87e7·16-s + (−6.99e6 − 6.99e6i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.965i)2-s + (1.20 − 1.20i)3-s + 0.864i·4-s + (−0.882 − 0.471i)5-s − 2.32·6-s + (0.574 + 0.574i)7-s + (−0.131 + 0.131i)8-s − 1.89i·9-s + (0.396 + 1.30i)10-s − 0.353·11-s + (1.03 + 1.03i)12-s + (0.636 − 0.636i)13-s − 1.10i·14-s + (−1.62 + 0.494i)15-s + 1.11·16-s + (−0.289 − 0.289i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0629i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.998 + 0.0629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.998 + 0.0629i$
Analytic conductor: \(4.56996\)
Root analytic conductor: \(2.13774\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :6),\ -0.998 + 0.0629i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0345427 - 1.09696i\)
\(L(\frac12)\) \(\approx\) \(0.0345427 - 1.09696i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.37e4 + 7.36e3i)T \)
good2 \( 1 + (61.7 + 61.7i)T + 4.09e3iT^{2} \)
3 \( 1 + (-877. + 877. i)T - 5.31e5iT^{2} \)
7 \( 1 + (-6.75e4 - 6.75e4i)T + 1.38e10iT^{2} \)
11 \( 1 + 6.26e5T + 3.13e12T^{2} \)
13 \( 1 + (-3.07e6 + 3.07e6i)T - 2.32e13iT^{2} \)
17 \( 1 + (6.99e6 + 6.99e6i)T + 5.82e14iT^{2} \)
19 \( 1 + 4.52e7iT - 2.21e15T^{2} \)
23 \( 1 + (-5.22e7 + 5.22e7i)T - 2.19e16iT^{2} \)
29 \( 1 + 5.40e8iT - 3.53e17T^{2} \)
31 \( 1 + 2.02e8T + 7.87e17T^{2} \)
37 \( 1 + (-2.57e9 - 2.57e9i)T + 6.58e18iT^{2} \)
41 \( 1 - 2.72e9T + 2.25e19T^{2} \)
43 \( 1 + (4.70e9 - 4.70e9i)T - 3.99e19iT^{2} \)
47 \( 1 + (-1.28e10 - 1.28e10i)T + 1.16e20iT^{2} \)
53 \( 1 + (-1.73e10 + 1.73e10i)T - 4.91e20iT^{2} \)
59 \( 1 - 1.93e10iT - 1.77e21T^{2} \)
61 \( 1 + 1.74e10T + 2.65e21T^{2} \)
67 \( 1 + (-7.91e10 - 7.91e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 1.08e11T + 1.64e22T^{2} \)
73 \( 1 + (3.78e10 - 3.78e10i)T - 2.29e22iT^{2} \)
79 \( 1 + 4.17e10iT - 5.90e22T^{2} \)
83 \( 1 + (-1.08e11 + 1.08e11i)T - 1.06e23iT^{2} \)
89 \( 1 - 6.78e11iT - 2.46e23T^{2} \)
97 \( 1 + (3.15e10 + 3.15e10i)T + 6.93e23iT^{2} \)
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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

−19.95070846611756816354274842803, −18.90468562075414538629839319403, −17.94898398005052347054966751493, −15.10113474696566515379818310471, −12.99756734134870370908977139080, −11.53207392955333400070089194008, −8.872445204143159001633741521473, −7.937025816709280483814592641887, −2.66241935376205989321161017992, −0.918409376795127847956385448676, 3.81986061633996826722945505417, 7.61773842319832447869724434945, 8.824306832211756052108533799759, 10.57856896451726461863657591596, 14.38053292985938444938607874920, 15.46160596231843517177474245525, 16.52358628153823926112770615112, 18.60095886441442903308135591827, 20.04612550192876578021662872217, 21.44797588983830973382595257427

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