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Properties

Label 2-5-5.2-c12-0-2
Degree $2$
Conductor $5$
Sign $0.999 + 0.0446i$
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  + (34.6 + 34.6i)2-s + (539. − 539. i)3-s − 1.68e3i·4-s + (1.29e4 + 8.80e3i)5-s + 3.74e4·6-s + (2.59e4 + 2.59e4i)7-s + (2.00e5 − 2.00e5i)8-s − 4.96e4i·9-s + (1.42e5 + 7.53e5i)10-s − 3.09e6·11-s + (−9.10e5 − 9.10e5i)12-s + (2.18e5 − 2.18e5i)13-s + 1.79e6i·14-s + (1.17e7 − 2.21e6i)15-s + 7.01e6·16-s + (−5.19e6 − 5.19e6i)17-s + ⋯
L(s)  = 1  + (0.542 + 0.542i)2-s + (0.739 − 0.739i)3-s − 0.412i·4-s + (0.826 + 0.563i)5-s + 0.801·6-s + (0.220 + 0.220i)7-s + (0.765 − 0.765i)8-s − 0.0934i·9-s + (0.142 + 0.753i)10-s − 1.74·11-s + (−0.304 − 0.304i)12-s + (0.0452 − 0.0452i)13-s + 0.238i·14-s + (1.02 − 0.194i)15-s + 0.417·16-s + (−0.215 − 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0446i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.999 + 0.0446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.999 + 0.0446i$
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: \((2,\ 5,\ (\ :6),\ 0.999 + 0.0446i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.54668 - 0.0569457i\)
\(L(\frac12)\) \(\approx\) \(2.54668 - 0.0569457i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.29e4 - 8.80e3i)T \)
good2 \( 1 + (-34.6 - 34.6i)T + 4.09e3iT^{2} \)
3 \( 1 + (-539. + 539. i)T - 5.31e5iT^{2} \)
7 \( 1 + (-2.59e4 - 2.59e4i)T + 1.38e10iT^{2} \)
11 \( 1 + 3.09e6T + 3.13e12T^{2} \)
13 \( 1 + (-2.18e5 + 2.18e5i)T - 2.32e13iT^{2} \)
17 \( 1 + (5.19e6 + 5.19e6i)T + 5.82e14iT^{2} \)
19 \( 1 - 4.05e7iT - 2.21e15T^{2} \)
23 \( 1 + (1.59e8 - 1.59e8i)T - 2.19e16iT^{2} \)
29 \( 1 + 8.83e8iT - 3.53e17T^{2} \)
31 \( 1 - 8.39e8T + 7.87e17T^{2} \)
37 \( 1 + (2.38e9 + 2.38e9i)T + 6.58e18iT^{2} \)
41 \( 1 - 3.94e9T + 2.25e19T^{2} \)
43 \( 1 + (-2.54e9 + 2.54e9i)T - 3.99e19iT^{2} \)
47 \( 1 + (-2.54e9 - 2.54e9i)T + 1.16e20iT^{2} \)
53 \( 1 + (8.59e8 - 8.59e8i)T - 4.91e20iT^{2} \)
59 \( 1 - 3.93e10iT - 1.77e21T^{2} \)
61 \( 1 - 4.84e10T + 2.65e21T^{2} \)
67 \( 1 + (4.84e10 + 4.84e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 7.15e10T + 1.64e22T^{2} \)
73 \( 1 + (-3.85e10 + 3.85e10i)T - 2.29e22iT^{2} \)
79 \( 1 + 5.14e10iT - 5.90e22T^{2} \)
83 \( 1 + (3.03e11 - 3.03e11i)T - 1.06e23iT^{2} \)
89 \( 1 + 5.82e11iT - 2.46e23T^{2} \)
97 \( 1 + (-3.48e11 - 3.48e11i)T + 6.93e23iT^{2} \)
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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

−21.09381555638808190060630407422, −19.19523277225996387298291287526, −18.11540059880151452199493998315, −15.62679497692445092579045736916, −14.10458887185281925422439377061, −13.27384290315460026992425211311, −10.24813640370536796506365414151, −7.63571855024723545752002536904, −5.67242890665179942033278325665, −2.16476082680819636060835882374, 2.64614412983406766514947750726, 4.70816182474164037004906422772, 8.457248597144657193926439775223, 10.40180966744711828180407884376, 12.75803637698671461821729296690, 14.02598161754478999503278145304, 15.98998276798011088847621612310, 17.75672369998011034146672458084, 20.33126884187733928892754928086, 20.92729481797513986628410183470

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