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Properties

Label 2-52-52.31-c3-0-11
Degree $2$
Conductor $52$
Sign $0.739 - 0.673i$
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  + (2.80 − 0.340i)2-s + 6.14i·3-s + (7.76 − 1.91i)4-s + (−1.41 + 1.41i)5-s + (2.09 + 17.2i)6-s + (−3.71 + 3.71i)7-s + (21.1 − 8.01i)8-s − 10.7·9-s + (−3.48 + 4.44i)10-s + (−1.79 + 1.79i)11-s + (11.7 + 47.7i)12-s + (−9.47 − 45.9i)13-s + (−9.15 + 11.6i)14-s + (−8.66 − 8.66i)15-s + (56.6 − 29.7i)16-s − 98.0i·17-s + ⋯
L(s)  = 1  + (0.992 − 0.120i)2-s + 1.18i·3-s + (0.971 − 0.239i)4-s + (−0.126 + 0.126i)5-s + (0.142 + 1.17i)6-s + (−0.200 + 0.200i)7-s + (0.935 − 0.354i)8-s − 0.396·9-s + (−0.110 + 0.140i)10-s + (−0.0491 + 0.0491i)11-s + (0.282 + 1.14i)12-s + (−0.202 − 0.979i)13-s + (−0.174 + 0.223i)14-s + (−0.149 − 0.149i)15-s + (0.885 − 0.464i)16-s − 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.739 - 0.673i$
Analytic conductor: \(3.06809\)
Root analytic conductor: \(1.75159\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :3/2),\ 0.739 - 0.673i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.15855 + 0.835969i\)
\(L(\frac12)\) \(\approx\) \(2.15855 + 0.835969i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.80 + 0.340i)T \)
13 \( 1 + (9.47 + 45.9i)T \)
good3 \( 1 - 6.14iT - 27T^{2} \)
5 \( 1 + (1.41 - 1.41i)T - 125iT^{2} \)
7 \( 1 + (3.71 - 3.71i)T - 343iT^{2} \)
11 \( 1 + (1.79 - 1.79i)T - 1.33e3iT^{2} \)
17 \( 1 + 98.0iT - 4.91e3T^{2} \)
19 \( 1 + (7.04 + 7.04i)T + 6.85e3iT^{2} \)
23 \( 1 + 135.T + 1.21e4T^{2} \)
29 \( 1 - 93.0T + 2.43e4T^{2} \)
31 \( 1 + (125. + 125. i)T + 2.97e4iT^{2} \)
37 \( 1 + (-50.2 - 50.2i)T + 5.06e4iT^{2} \)
41 \( 1 + (187. - 187. i)T - 6.89e4iT^{2} \)
43 \( 1 - 426.T + 7.95e4T^{2} \)
47 \( 1 + (121. - 121. i)T - 1.03e5iT^{2} \)
53 \( 1 - 418.T + 1.48e5T^{2} \)
59 \( 1 + (236. - 236. i)T - 2.05e5iT^{2} \)
61 \( 1 + 507.T + 2.26e5T^{2} \)
67 \( 1 + (47.6 + 47.6i)T + 3.00e5iT^{2} \)
71 \( 1 + (820. + 820. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-711. - 711. i)T + 3.89e5iT^{2} \)
79 \( 1 - 573. iT - 4.93e5T^{2} \)
83 \( 1 + (557. + 557. i)T + 5.71e5iT^{2} \)
89 \( 1 + (893. + 893. i)T + 7.04e5iT^{2} \)
97 \( 1 + (61.6 - 61.6i)T - 9.12e5iT^{2} \)
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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

−15.22836309505504936112119267853, −14.10744654906098555863741141788, −12.85761879124842060964598794581, −11.61336705228439902657519221991, −10.51401822793959487621250980503, −9.516842585292993363682116686004, −7.48254918773913077377809069841, −5.69871598261613549800119653773, −4.47944723494322453012097874948, −3.04735720444823422292687746914, 1.93896343956732933546222882863, 4.11193413547961186821423358474, 6.06981553417320255690793929981, 7.04750856785074737680010478834, 8.260685371794923998746714052545, 10.44313325746849201145941639303, 11.96291005586754733844325166311, 12.56423637842864807615513979969, 13.63079563751216578795232765999, 14.44407234014457884090880651342

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