Properties

Label 2-52-52.3-c2-0-3
Degree $2$
Conductor $52$
Sign $-0.194 - 0.980i$
Analytic cond. $1.41689$
Root an. cond. $1.19033$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.39 + 1.43i)2-s + (−3.19 + 1.84i)3-s + (−0.131 + 3.99i)4-s + 2.13·5-s + (−7.09 − 2.02i)6-s + (1.80 + 1.03i)7-s + (−5.92 + 5.37i)8-s + (2.31 − 4.01i)9-s + (2.96 + 3.06i)10-s + (14.3 − 8.29i)11-s + (−6.95 − 13.0i)12-s + (6.93 − 10.9i)13-s + (1.01 + 4.03i)14-s + (−6.81 + 3.93i)15-s + (−15.9 − 1.05i)16-s + (1.97 − 3.42i)17-s + ⋯
L(s)  = 1  + (0.695 + 0.718i)2-s + (−1.06 + 0.615i)3-s + (−0.0329 + 0.999i)4-s + 0.426·5-s + (−1.18 − 0.338i)6-s + (0.257 + 0.148i)7-s + (−0.741 + 0.671i)8-s + (0.257 − 0.445i)9-s + (0.296 + 0.306i)10-s + (1.30 − 0.754i)11-s + (−0.579 − 1.08i)12-s + (0.533 − 0.845i)13-s + (0.0721 + 0.288i)14-s + (−0.454 + 0.262i)15-s + (−0.997 − 0.0659i)16-s + (0.116 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $-0.194 - 0.980i$
Analytic conductor: \(1.41689\)
Root analytic conductor: \(1.19033\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1),\ -0.194 - 0.980i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.803925 + 0.979146i\)
\(L(\frac12)\) \(\approx\) \(0.803925 + 0.979146i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 1.43i)T \)
13 \( 1 + (-6.93 + 10.9i)T \)
good3 \( 1 + (3.19 - 1.84i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 - 2.13T + 25T^{2} \)
7 \( 1 + (-1.80 - 1.03i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-14.3 + 8.29i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (-1.97 + 3.42i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-11.8 - 6.83i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (32.3 - 18.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.0 - 19.2i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 47.2iT - 961T^{2} \)
37 \( 1 + (11.7 + 20.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-31.2 - 54.0i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-12.3 - 7.15i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 19.4iT - 2.20e3T^{2} \)
53 \( 1 - 30.8T + 2.80e3T^{2} \)
59 \( 1 + (48.5 + 28.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-36.7 + 63.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (63.3 - 36.6i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-73.6 - 42.5i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 83.3T + 5.32e3T^{2} \)
79 \( 1 + 6.78iT - 6.24e3T^{2} \)
83 \( 1 - 80.5iT - 6.88e3T^{2} \)
89 \( 1 + (-9.66 - 16.7i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-4.83 + 8.37i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−15.75234436710207064403791739355, −14.42889402035770645993400563395, −13.49428599567667361322223741567, −11.96156659960526100744755456966, −11.25798493692863861965628251230, −9.646002942917567311299033163612, −8.044307935646613387550315210898, −6.16159968895469801365635771308, −5.52014698395570730201087539233, −3.87702736687391797448050563717, 1.54741425026018384549030208597, 4.27792310882883191848878172875, 5.90006471646106149390448299376, 6.80419686752494141848813141219, 9.262878932735929101831664904492, 10.58245359956628474300672303504, 11.86163555199424379551665438102, 12.18301814346312959689117073076, 13.68695848088841680497480871243, 14.44754285617562155589878514612

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