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Properties

Label 2-52-52.47-c3-0-8
Degree $2$
Conductor $52$
Sign $0.376 - 0.926i$
Analytic cond. $3.06809$
Root an. cond. $1.75159$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.81 + 0.289i)2-s + 7.82i·3-s + (7.83 + 1.62i)4-s + (−8.33 − 8.33i)5-s + (−2.26 + 22.0i)6-s + (11.3 + 11.3i)7-s + (21.5 + 6.84i)8-s − 34.3·9-s + (−21.0 − 25.8i)10-s + (−25.6 − 25.6i)11-s + (−12.7 + 61.3i)12-s + (46.5 − 5.26i)13-s + (28.5 + 35.1i)14-s + (65.2 − 65.2i)15-s + (58.7 + 25.4i)16-s − 78.9i·17-s + ⋯
L(s)  = 1  + (0.994 + 0.102i)2-s + 1.50i·3-s + (0.979 + 0.203i)4-s + (−0.745 − 0.745i)5-s + (−0.154 + 1.49i)6-s + (0.611 + 0.611i)7-s + (0.953 + 0.302i)8-s − 1.27·9-s + (−0.665 − 0.817i)10-s + (−0.702 − 0.702i)11-s + (−0.306 + 1.47i)12-s + (0.993 − 0.112i)13-s + (0.545 + 0.670i)14-s + (1.12 − 1.12i)15-s + (0.917 + 0.398i)16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(3.06809\)
Root analytic conductor: \(1.75159\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :3/2),\ 0.376 - 0.926i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.86686 + 1.25615i\)
\(L(\frac12)\) \(\approx\) \(1.86686 + 1.25615i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.81 - 0.289i)T \)
13 \( 1 + (-46.5 + 5.26i)T \)
good3 \( 1 - 7.82iT - 27T^{2} \)
5 \( 1 + (8.33 + 8.33i)T + 125iT^{2} \)
7 \( 1 + (-11.3 - 11.3i)T + 343iT^{2} \)
11 \( 1 + (25.6 + 25.6i)T + 1.33e3iT^{2} \)
17 \( 1 + 78.9iT - 4.91e3T^{2} \)
19 \( 1 + (-40.6 + 40.6i)T - 6.85e3iT^{2} \)
23 \( 1 + 16.5T + 1.21e4T^{2} \)
29 \( 1 + 273.T + 2.43e4T^{2} \)
31 \( 1 + (166. - 166. i)T - 2.97e4iT^{2} \)
37 \( 1 + (-88.3 + 88.3i)T - 5.06e4iT^{2} \)
41 \( 1 + (-341. - 341. i)T + 6.89e4iT^{2} \)
43 \( 1 + 381.T + 7.95e4T^{2} \)
47 \( 1 + (-251. - 251. i)T + 1.03e5iT^{2} \)
53 \( 1 + 400.T + 1.48e5T^{2} \)
59 \( 1 + (449. + 449. i)T + 2.05e5iT^{2} \)
61 \( 1 - 157.T + 2.26e5T^{2} \)
67 \( 1 + (138. - 138. i)T - 3.00e5iT^{2} \)
71 \( 1 + (-236. + 236. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-59.1 + 59.1i)T - 3.89e5iT^{2} \)
79 \( 1 - 559. iT - 4.93e5T^{2} \)
83 \( 1 + (-140. + 140. i)T - 5.71e5iT^{2} \)
89 \( 1 + (-137. + 137. i)T - 7.04e5iT^{2} \)
97 \( 1 + (-598. - 598. i)T + 9.12e5iT^{2} \)
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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

−15.38447695720384565755294147509, −14.27794179073508350787998041124, −12.94693149271694460386202671725, −11.49178847920105229404833872393, −10.96463748176286396612878907977, −9.161742271669840738234837632333, −7.932284507086089469816687746548, −5.53870885522283759298087477430, −4.69638201380513655600393529198, −3.35895576359499751242565570999, 1.80310413986529452941682273873, 3.80216458225220400194478722148, 5.93455458478931851247574180806, 7.32407644201237116951755191245, 7.78173392220828896594158953796, 10.70442294873512718317237766368, 11.50187031696891753050080233956, 12.67365722498515995505738504015, 13.44264811335915391926024399722, 14.52086379267175774581881162008

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