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Properties

Label 2-52-52.23-c2-0-11
Degree $2$
Conductor $52$
Sign $0.193 + 0.981i$
Analytic cond. $1.41689$
Root an. cond. $1.19033$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.359 − 1.96i)2-s + (4.31 − 2.49i)3-s + (−3.74 − 1.41i)4-s + 4.86i·5-s + (−3.34 − 9.38i)6-s + (−5.11 + 8.86i)7-s + (−4.13 + 6.84i)8-s + (7.92 − 13.7i)9-s + (9.56 + 1.74i)10-s + (−2.41 − 4.17i)11-s + (−19.6 + 3.20i)12-s + (−0.594 − 12.9i)13-s + (15.5 + 13.2i)14-s + (12.1 + 20.9i)15-s + (11.9 + 10.5i)16-s + (−2.15 + 3.72i)17-s + ⋯
L(s)  = 1  + (0.179 − 0.983i)2-s + (1.43 − 0.830i)3-s + (−0.935 − 0.354i)4-s + 0.972i·5-s + (−0.558 − 1.56i)6-s + (−0.730 + 1.26i)7-s + (−0.516 + 0.856i)8-s + (0.880 − 1.52i)9-s + (0.956 + 0.174i)10-s + (−0.219 − 0.379i)11-s + (−1.63 + 0.267i)12-s + (−0.0456 − 0.998i)13-s + (1.11 + 0.946i)14-s + (0.807 + 1.39i)15-s + (0.749 + 0.662i)16-s + (−0.126 + 0.219i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.193 + 0.981i$
Analytic conductor: \(1.41689\)
Root analytic conductor: \(1.19033\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1),\ 0.193 + 0.981i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.20331 - 0.989306i\)
\(L(\frac12)\) \(\approx\) \(1.20331 - 0.989306i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.359 + 1.96i)T \)
13 \( 1 + (0.594 + 12.9i)T \)
good3 \( 1 + (-4.31 + 2.49i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 - 4.86iT - 25T^{2} \)
7 \( 1 + (5.11 - 8.86i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (2.41 + 4.17i)T + (-60.5 + 104. i)T^{2} \)
17 \( 1 + (2.15 - 3.72i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-3.60 + 6.24i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (14.9 - 8.61i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (18.5 + 32.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 28.8T + 961T^{2} \)
37 \( 1 + (-15.8 + 9.12i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (51.4 - 29.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-36.0 - 20.8i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 28.2T + 2.20e3T^{2} \)
53 \( 1 - 35.8T + 2.80e3T^{2} \)
59 \( 1 + (14.3 - 24.8i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.4 - 57.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-34.9 - 60.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-13.5 + 23.4i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + 108. iT - 5.32e3T^{2} \)
79 \( 1 + 78.1iT - 6.24e3T^{2} \)
83 \( 1 + 67.1T + 6.88e3T^{2} \)
89 \( 1 + (-51.6 + 29.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-28.5 - 16.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−14.76701164866279146702730994663, −13.58394446816281542658727750751, −12.89283309751898573651541115040, −11.74261292357817774791175541325, −10.13349891055959867721029553060, −9.021228792693345584830896060940, −7.917683099550775110551799389311, −6.08125743651934224472740701609, −3.22763134421132771307791371346, −2.48606144058433279010547597398, 3.73468504699368836628174148401, 4.70440582618010530581982332177, 7.01476550137779735465313842015, 8.279001578793850611570529922907, 9.279889343054268910277109725489, 10.09251659151095551518199914336, 12.65567183204474317988968758783, 13.68894382212617929819133592274, 14.26655877094305344580312427235, 15.55462936793073014532425425388

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