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Properties

Label 2-52-52.19-c1-0-1
Degree $2$
Conductor $52$
Sign $0.477 - 0.878i$
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  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (0.633 − 0.633i)5-s + (−2 − 1.99i)8-s + (−1.5 − 2.59i)9-s + (1.09 + 0.633i)10-s + (3.59 − 0.232i)13-s + (1.99 − 3.46i)16-s + (−6.86 + 3.96i)17-s + (3 − 3i)18-s + (−0.464 + 1.73i)20-s + 4.19i·25-s + (1.63 + 4.83i)26-s + (−3.33 + 5.76i)29-s + (5.46 + 1.46i)32-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + (0.283 − 0.283i)5-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)9-s + (0.347 + 0.200i)10-s + (0.997 − 0.0643i)13-s + (0.499 − 0.866i)16-s + (−1.66 + 0.961i)17-s + (0.707 − 0.707i)18-s + (−0.103 + 0.387i)20-s + 0.839i·25-s + (0.320 + 0.947i)26-s + (−0.618 + 1.07i)29-s + (0.965 + 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757964 + 0.450741i\)
\(L(\frac12)\) \(\approx\) \(0.757964 + 0.450741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
13 \( 1 + (-3.59 + 0.232i)T \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.633 + 0.633i)T - 5iT^{2} \)
7 \( 1 + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (6.86 - 3.96i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.33 - 5.76i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 + (-11.6 + 3.13i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.66 + 9.96i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.69 + 4.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-9.83 - 9.83i)T + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (4.09 - 1.09i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.83 + 1.83i)T + (84.0 + 48.5i)T^{2} \)
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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

−15.54878620892821321430718376387, −14.74989370126657612733207381925, −13.48281689196067280512821285553, −12.68520356602253402365644388898, −11.12057149536826258913955963587, −9.261965981035636758590460985983, −8.486054835856890677227479457762, −6.75443969258722335577912003593, −5.66641055851929360058606293608, −3.87521082344718984008305960188, 2.52657693685617399564981088391, 4.53358389757030762429745569301, 6.15454646270486005328547451229, 8.297346407402089548407318600183, 9.571858930860050498131620442129, 10.91328134438960859545061080082, 11.54926806610758296882055014005, 13.28050546797359662800217382804, 13.71138957338138956792450886787, 15.03459703779609310404345655658

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