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Properties

Label 2-252-63.16-c1-0-1
Degree $2$
Conductor $252$
Sign $0.520 - 0.854i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
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  + (1.13 + 1.31i)3-s − 1.52·5-s + (2.53 + 0.752i)7-s + (−0.440 + 2.96i)9-s + 0.835·11-s + (1.81 + 3.13i)13-s + (−1.73 − 2.00i)15-s + (0.301 + 0.521i)17-s + (0.846 − 1.46i)19-s + (1.88 + 4.17i)21-s − 6.14·23-s − 2.66·25-s + (−4.39 + 2.77i)27-s + (4.99 − 8.65i)29-s + (1.65 − 2.86i)31-s + ⋯
L(s)  = 1  + (0.653 + 0.757i)3-s − 0.683·5-s + (0.958 + 0.284i)7-s + (−0.146 + 0.989i)9-s + 0.251·11-s + (0.502 + 0.870i)13-s + (−0.446 − 0.517i)15-s + (0.0730 + 0.126i)17-s + (0.194 − 0.336i)19-s + (0.410 + 0.911i)21-s − 1.28·23-s − 0.532·25-s + (−0.845 + 0.534i)27-s + (0.927 − 1.60i)29-s + (0.297 − 0.514i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.520 - 0.854i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.520 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29705 + 0.728790i\)
\(L(\frac12)\) \(\approx\) \(1.29705 + 0.728790i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.13 - 1.31i)T \)
7 \( 1 + (-2.53 - 0.752i)T \)
good5 \( 1 + 1.52T + 5T^{2} \)
11 \( 1 - 0.835T + 11T^{2} \)
13 \( 1 + (-1.81 - 3.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.301 - 0.521i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.846 + 1.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.14T + 23T^{2} \)
29 \( 1 + (-4.99 + 8.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.65 + 2.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.39 + 7.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.51 - 6.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.846 + 1.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.23 + 7.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.99 + 6.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0652 - 0.112i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.38 - 4.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.12 + 1.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.39T + 71T^{2} \)
73 \( 1 + (2.25 + 3.90i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.87 + 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.16 - 5.47i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.531 - 0.920i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.76 - 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.73966130531793188493065294570, −11.47083816256373407495926119164, −10.22713974293107871124655631715, −9.244109210256993023702287958143, −8.275052641324200979630152090172, −7.68254436387148205725937174567, −6.02611823749599489196347486295, −4.56480515553907284700773300125, −3.89316801158867157903300132888, −2.19468336418525194293887811872, 1.35333897399538017759555762111, 3.12414426932274817821755248041, 4.34502902705142054021589649419, 5.90629752829256509171507308166, 7.20153736864324195789717513307, 8.042255120339503118883332707601, 8.553186882198115376319576499999, 9.970147888481140400426869696025, 11.12241274750302212770138852946, 11.99065419935889401010638388151

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