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Properties

Label 2-52-13.4-c1-0-0
Degree $2$
Conductor $52$
Sign $0.964 + 0.265i$
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  + (0.5 − 0.866i)3-s + (−1.5 + 0.866i)7-s + (1 + 1.73i)9-s + (−4.5 − 2.59i)11-s + (−1 + 3.46i)13-s + (−1.5 − 2.59i)17-s + (4.5 − 2.59i)19-s + 1.73i·21-s + (−1.5 + 2.59i)23-s + 5·25-s + 5·27-s + (4.5 − 7.79i)29-s + 3.46i·31-s + (−4.5 + 2.59i)33-s + (−4.5 − 2.59i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.566 + 0.327i)7-s + (0.333 + 0.577i)9-s + (−1.35 − 0.783i)11-s + (−0.277 + 0.960i)13-s + (−0.363 − 0.630i)17-s + (1.03 − 0.596i)19-s + 0.377i·21-s + (−0.312 + 0.541i)23-s + 25-s + 0.962·27-s + (0.835 − 1.44i)29-s + 0.622i·31-s + (−0.783 + 0.452i)33-s + (−0.739 − 0.427i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.850412 - 0.114733i\)
\(L(\frac12)\) \(\approx\) \(0.850412 - 0.114733i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (1 - 3.46i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + (4.5 + 2.59i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-4.5 + 2.59i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.5 + 2.59i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (-13.5 - 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.5 - 6.06i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.74291947372379300700983587118, −13.93285200414199036646100619717, −13.38425799277543966618613842522, −12.13791517728018180950124500331, −10.79723789966663697352239370105, −9.423342291368123136068399881997, −8.055540714039643496125272961652, −6.83673594255509786305448272073, −5.09523319457513758867591491269, −2.71508966024172863848962411792, 3.24703217447323160806390234742, 5.02036750892239134221500152048, 6.87974091552698571658712271357, 8.309031643170444544219934443170, 9.908048957039052013507580469523, 10.43147073634040594561024974177, 12.36680837687021442250346416085, 13.12585366359457438970843965620, 14.65290280526912027259907662418, 15.50720538951156103155418630581

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