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Properties

Label 2-52-52.31-c1-0-2
Degree $2$
Conductor $52$
Sign $0.822 + 0.568i$
Analytic cond. $0.415222$
Root an. cond. $0.644377$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 + 0.332i)2-s − 1.47i·3-s + (1.77 − 0.914i)4-s + (0.707 − 0.707i)5-s + (0.490 + 2.02i)6-s + (1.04 − 1.04i)7-s + (−2.14 + 1.84i)8-s + 0.828·9-s + (−0.736 + 1.20i)10-s + (−3.55 + 3.55i)11-s + (−1.34 − 2.62i)12-s + (−3.53 + 0.707i)13-s + (−1.08 + 1.77i)14-s + (−1.04 − 1.04i)15-s + (2.32 − 3.25i)16-s − 0.171i·17-s + ⋯
L(s)  = 1  + (−0.971 + 0.235i)2-s − 0.850i·3-s + (0.889 − 0.457i)4-s + (0.316 − 0.316i)5-s + (0.200 + 0.826i)6-s + (0.393 − 0.393i)7-s + (−0.756 + 0.653i)8-s + 0.276·9-s + (−0.233 + 0.381i)10-s + (−1.07 + 1.07i)11-s + (−0.388 − 0.756i)12-s + (−0.980 + 0.196i)13-s + (−0.290 + 0.475i)14-s + (−0.269 − 0.269i)15-s + (0.582 − 0.813i)16-s − 0.0416i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.822 + 0.568i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1/2),\ 0.822 + 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.580337 - 0.181002i\)
\(L(\frac12)\) \(\approx\) \(0.580337 - 0.181002i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.332i)T \)
13 \( 1 + (3.53 - 0.707i)T \)
good3 \( 1 + 1.47iT - 3T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T - 5iT^{2} \)
7 \( 1 + (-1.04 + 1.04i)T - 7iT^{2} \)
11 \( 1 + (3.55 - 3.55i)T - 11iT^{2} \)
17 \( 1 + 0.171iT - 17T^{2} \)
19 \( 1 + (-5.03 - 5.03i)T + 19iT^{2} \)
23 \( 1 - 2.08T + 23T^{2} \)
29 \( 1 + 3.41T + 29T^{2} \)
31 \( 1 + (2.08 + 2.08i)T + 31iT^{2} \)
37 \( 1 + (6.12 + 6.12i)T + 37iT^{2} \)
41 \( 1 + (-4.24 + 4.24i)T - 41iT^{2} \)
43 \( 1 + 5.64T + 43T^{2} \)
47 \( 1 + (1.04 - 1.04i)T - 47iT^{2} \)
53 \( 1 - 8.24T + 53T^{2} \)
59 \( 1 + (-7.11 + 7.11i)T - 59iT^{2} \)
61 \( 1 + 4.82T + 61T^{2} \)
67 \( 1 + (1.47 + 1.47i)T + 67iT^{2} \)
71 \( 1 + (9.02 + 9.02i)T + 71iT^{2} \)
73 \( 1 + (-1.65 - 1.65i)T + 73iT^{2} \)
79 \( 1 - 0.863iT - 79T^{2} \)
83 \( 1 + (0.610 + 0.610i)T + 83iT^{2} \)
89 \( 1 + (2.24 + 2.24i)T + 89iT^{2} \)
97 \( 1 + (2.41 - 2.41i)T - 97iT^{2} \)
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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

−15.54634689402109365929981718497, −14.36706003906047866157773298731, −12.94980312721025985092375804476, −11.97162463481887189292741635736, −10.41068669110292873539074336885, −9.454981696008928532702650124557, −7.67612274844376929891940654308, −7.25266987496594696569803279340, −5.33902559114650045858671359001, −1.86921445022426664662578267791, 2.90441970556398063376289308610, 5.27537125655945897425751179119, 7.21484705843595877106643089991, 8.617027492698527973125131110966, 9.818665056297153844284809077893, 10.63715670884158388629471046477, 11.70654316803090544953144988970, 13.25124367323121791021311445650, 14.91049644126229172027590644880, 15.77169893888192543984164944328

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