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Properties

Label 2-5-5.2-c12-0-1
Degree $2$
Conductor $5$
Sign $-0.592 - 0.805i$
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  + (77.8 + 77.8i)2-s + (−451. + 451. i)3-s + 8.03e3i·4-s + (−1.25e3 − 1.55e4i)5-s − 7.02e4·6-s + (1.07e5 + 1.07e5i)7-s + (−3.06e5 + 3.06e5i)8-s + 1.24e5i·9-s + (1.11e6 − 1.31e6i)10-s + 7.06e5·11-s + (−3.62e6 − 3.62e6i)12-s + (4.65e6 − 4.65e6i)13-s + 1.67e7i·14-s + (7.59e6 + 6.45e6i)15-s − 1.48e7·16-s + (−2.26e7 − 2.26e7i)17-s + ⋯
L(s)  = 1  + (1.21 + 1.21i)2-s + (−0.618 + 0.618i)3-s + 1.96i·4-s + (−0.0804 − 0.996i)5-s − 1.50·6-s + (0.914 + 0.914i)7-s + (−1.17 + 1.17i)8-s + 0.234i·9-s + (1.11 − 1.31i)10-s + 0.398·11-s + (−1.21 − 1.21i)12-s + (0.965 − 0.965i)13-s + 2.22i·14-s + (0.666 + 0.566i)15-s − 0.887·16-s + (−0.939 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.592 - 0.805i$
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: \((2,\ 5,\ (\ :6),\ -0.592 - 0.805i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(1.07146 + 2.11797i\)
\(L(\frac12)\) \(\approx\) \(1.07146 + 2.11797i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.25e3 + 1.55e4i)T \)
good2 \( 1 + (-77.8 - 77.8i)T + 4.09e3iT^{2} \)
3 \( 1 + (451. - 451. i)T - 5.31e5iT^{2} \)
7 \( 1 + (-1.07e5 - 1.07e5i)T + 1.38e10iT^{2} \)
11 \( 1 - 7.06e5T + 3.13e12T^{2} \)
13 \( 1 + (-4.65e6 + 4.65e6i)T - 2.32e13iT^{2} \)
17 \( 1 + (2.26e7 + 2.26e7i)T + 5.82e14iT^{2} \)
19 \( 1 - 7.98e6iT - 2.21e15T^{2} \)
23 \( 1 + (-7.99e6 + 7.99e6i)T - 2.19e16iT^{2} \)
29 \( 1 + 2.39e8iT - 3.53e17T^{2} \)
31 \( 1 - 5.67e8T + 7.87e17T^{2} \)
37 \( 1 + (-4.93e8 - 4.93e8i)T + 6.58e18iT^{2} \)
41 \( 1 + 2.49e9T + 2.25e19T^{2} \)
43 \( 1 + (3.60e8 - 3.60e8i)T - 3.99e19iT^{2} \)
47 \( 1 + (4.12e9 + 4.12e9i)T + 1.16e20iT^{2} \)
53 \( 1 + (3.21e9 - 3.21e9i)T - 4.91e20iT^{2} \)
59 \( 1 - 2.34e10iT - 1.77e21T^{2} \)
61 \( 1 - 7.36e10T + 2.65e21T^{2} \)
67 \( 1 + (9.72e10 + 9.72e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 3.89e10T + 1.64e22T^{2} \)
73 \( 1 + (1.90e11 - 1.90e11i)T - 2.29e22iT^{2} \)
79 \( 1 - 4.54e10iT - 5.90e22T^{2} \)
83 \( 1 + (-3.66e11 + 3.66e11i)T - 1.06e23iT^{2} \)
89 \( 1 + 2.98e10iT - 2.46e23T^{2} \)
97 \( 1 + (3.55e11 + 3.55e11i)T + 6.93e23iT^{2} \)
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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

−21.95059728816609346067960956733, −20.78537421111291841510921689267, −17.57881936024702342091738634807, −16.20270078959681897591750960171, −15.30462480748997884608368340710, −13.41996806682061294061125463231, −11.71552402310467238178045924645, −8.338165102014442710893053657527, −5.65400380903218048322695848882, −4.60422025783223701540472448560, 1.51319891442807414867368131231, 3.99578611739623575980616146656, 6.49617540190275593020515151230, 10.82765337713340485306515508727, 11.64121234655530604303251254723, 13.46802679501182408594210834409, 14.68215997561943609575250691410, 17.69948587034749138675591980916, 19.18733378123963058746361718080, 20.73577308656506636290877160167

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