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Properties

Label 2-22e2-11.3-c1-0-6
Degree $2$
Conductor $484$
Sign $0.0694 + 0.997i$
Analytic cond. $3.86475$
Root an. cond. $1.96589$
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  + (−0.118 − 0.363i)3-s + (−0.5 − 0.363i)5-s + (0.881 − 2.71i)7-s + (2.30 − 1.67i)9-s + (−3.73 + 2.71i)13-s + (−0.0729 + 0.224i)15-s + (−3.73 − 2.71i)17-s + (−0.881 − 2.71i)19-s − 1.09·21-s + 6.47·23-s + (−1.42 − 4.39i)25-s + (−1.80 − 1.31i)27-s + (1.97 − 6.06i)29-s + (4.73 − 3.44i)31-s + (−1.42 + 1.03i)35-s + ⋯
L(s)  = 1  + (−0.0681 − 0.209i)3-s + (−0.223 − 0.162i)5-s + (0.333 − 1.02i)7-s + (0.769 − 0.559i)9-s + (−1.03 + 0.752i)13-s + (−0.0188 + 0.0579i)15-s + (−0.906 − 0.658i)17-s + (−0.202 − 0.622i)19-s − 0.237·21-s + 1.34·23-s + (−0.285 − 0.878i)25-s + (−0.348 − 0.252i)27-s + (0.366 − 1.12i)29-s + (0.850 − 0.618i)31-s + (−0.241 + 0.175i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(484\)    =    \(2^{2} \cdot 11^{2}\)
Sign: $0.0694 + 0.997i$
Analytic conductor: \(3.86475\)
Root analytic conductor: \(1.96589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{484} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 484,\ (\ :1/2),\ 0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922226 - 0.860230i\)
\(L(\frac12)\) \(\approx\) \(0.922226 - 0.860230i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.118 + 0.363i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.5 + 0.363i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.881 + 2.71i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.73 - 2.71i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.73 + 2.71i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.881 + 2.71i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 + (-1.97 + 6.06i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.73 + 3.44i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.5 + 4.61i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.97 - 6.06i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-1.11 - 3.44i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.73 - 4.16i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.11 - 9.59i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.73 + 2.71i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 4.94T + 67T^{2} \)
71 \( 1 + (-6.97 - 5.06i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.73 - 11.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.2 + 8.14i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.9 - 9.42i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 + (4.5 - 3.26i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.85390155896559336979287639436, −9.792273899786752086548286743161, −9.146068480775346986019432710860, −7.82713806560578320444658291724, −7.11832936344702484229128576700, −6.40004923961776677319836537580, −4.60336868034825221588603688852, −4.30871932589542957084370356967, −2.51802703799264709318050880094, −0.801613318190151122262043068152, 1.89877732161946509482565963949, 3.19390497325633154965789678684, 4.69357208466460970613038506648, 5.32802227895148200590078060467, 6.64106138948068699748787097535, 7.61230725115201762146920952303, 8.503092639888024761302879940723, 9.408912645636780409312259592361, 10.43988622905152310520165892294, 11.02589939538719343387928094768

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