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Properties

Label 2-52-52.19-c1-0-4
Degree $2$
Conductor $52$
Sign $-0.243 + 0.969i$
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  + (0.0921 − 1.41i)2-s + (−1.81 − 1.04i)3-s + (−1.98 − 0.260i)4-s + (0.894 − 0.894i)5-s + (−1.64 + 2.46i)6-s + (4.37 + 1.17i)7-s + (−0.549 + 2.77i)8-s + (0.693 + 1.20i)9-s + (−1.17 − 1.34i)10-s + (−0.404 − 1.51i)11-s + (3.32 + 2.54i)12-s + (−2.03 + 2.97i)13-s + (2.05 − 6.06i)14-s + (−2.55 + 0.685i)15-s + (3.86 + 1.03i)16-s + (−0.0484 + 0.0279i)17-s + ⋯
L(s)  = 1  + (0.0651 − 0.997i)2-s + (−1.04 − 0.604i)3-s + (−0.991 − 0.130i)4-s + (0.399 − 0.399i)5-s + (−0.671 + 1.00i)6-s + (1.65 + 0.442i)7-s + (−0.194 + 0.980i)8-s + (0.231 + 0.400i)9-s + (−0.372 − 0.425i)10-s + (−0.122 − 0.455i)11-s + (0.959 + 0.735i)12-s + (−0.565 + 0.824i)13-s + (0.549 − 1.61i)14-s + (−0.660 + 0.176i)15-s + (0.966 + 0.257i)16-s + (−0.0117 + 0.00678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $-0.243 + 0.969i$
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: \((2,\ 52,\ (\ :1/2),\ -0.243 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424998 - 0.545154i\)
\(L(\frac12)\) \(\approx\) \(0.424998 - 0.545154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0921 + 1.41i)T \)
13 \( 1 + (2.03 - 2.97i)T \)
good3 \( 1 + (1.81 + 1.04i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.894 + 0.894i)T - 5iT^{2} \)
7 \( 1 + (-4.37 - 1.17i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.404 + 1.51i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.0484 - 0.0279i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.576 - 2.15i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.528 - 0.916i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.67 + 6.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.52 + 4.52i)T + 31iT^{2} \)
37 \( 1 + (1.86 - 0.5i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.401 - 1.5i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.04 - 1.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.38 - 5.38i)T - 47iT^{2} \)
53 \( 1 + 4.40T + 53T^{2} \)
59 \( 1 + (8.67 + 2.32i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.00 - 1.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.86 + 2.37i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.76 + 6.59i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.45 + 5.45i)T + 73iT^{2} \)
79 \( 1 + 8.87iT - 79T^{2} \)
83 \( 1 + (-6.96 - 6.96i)T + 83iT^{2} \)
89 \( 1 + (15.3 - 4.10i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.440 + 0.118i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.79687556700536264610536400692, −13.77104903542041444907563120508, −12.49825298751140775535912827127, −11.64086547082700748343022207661, −11.02904579139956961548916061486, −9.358021224151850392706150250478, −7.992785898895246225670855774676, −5.81334217767118909951882546782, −4.74750394712209040073530888892, −1.71285655550302361469242236967, 4.65233643092870854138479557057, 5.40339185593007627962314478852, 7.02573671932970479975463391160, 8.330986230508667784957644520144, 10.12722190839460545423448356788, 10.93663963838438387540328554010, 12.40296510331885942119516875727, 14.02269120783984849641037885485, 14.76214536506127962630172511756, 15.85234760179041510366532707275

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