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Properties

Label 2-252-9.7-c1-0-3
Degree $2$
Conductor $252$
Sign $0.998 + 0.0576i$
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  + (1.64 − 0.545i)3-s + (0.849 + 1.47i)5-s + (0.5 − 0.866i)7-s + (2.40 − 1.79i)9-s + (−1.23 + 2.14i)11-s + (−0.388 − 0.673i)13-s + (2.19 + 1.95i)15-s + 2.81·17-s − 4.98·19-s + (0.349 − 1.69i)21-s + (−0.356 − 0.616i)23-s + (1.05 − 1.82i)25-s + (2.97 − 4.25i)27-s + (−2.25 + 3.90i)29-s + (−2.54 − 4.41i)31-s + ⋯
L(s)  = 1  + (0.949 − 0.314i)3-s + (0.380 + 0.658i)5-s + (0.188 − 0.327i)7-s + (0.801 − 0.597i)9-s + (−0.373 + 0.646i)11-s + (−0.107 − 0.186i)13-s + (0.567 + 0.505i)15-s + 0.681·17-s − 1.14·19-s + (0.0763 − 0.370i)21-s + (−0.0742 − 0.128i)23-s + (0.211 − 0.365i)25-s + (0.572 − 0.819i)27-s + (−0.418 + 0.725i)29-s + (−0.457 − 0.793i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.998 + 0.0576i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.998 + 0.0576i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74933 - 0.0504494i\)
\(L(\frac12)\) \(\approx\) \(1.74933 - 0.0504494i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 0.545i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.849 - 1.47i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.23 - 2.14i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.388 + 0.673i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.81T + 17T^{2} \)
19 \( 1 + 4.98T + 19T^{2} \)
23 \( 1 + (0.356 + 0.616i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.25 - 3.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 + 4.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.87T + 37T^{2} \)
41 \( 1 + (-2.93 - 5.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.32 + 4.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.49 - 11.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.88T + 53T^{2} \)
59 \( 1 + (7.14 + 12.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.15 - 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.99 + 6.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 4.98T + 73T^{2} \)
79 \( 1 + (-4.60 + 7.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.40 - 7.63i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.65T + 89T^{2} \)
97 \( 1 + (4.32 - 7.48i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.32204655448213822019311200786, −10.83032114973578574786466118747, −10.12460296603982613872253832124, −9.155258042267239360438596881160, −7.999846285896156494990053914020, −7.23418128840293447553933237774, −6.19569580088585289446433324955, −4.54728175548612171116546703887, −3.17765900896136957523199323648, −1.94969095509039845999500024896, 1.89681628225916855404210113582, 3.36413129489700293149825330547, 4.71448675895617768107620907995, 5.77916325892318407706627560349, 7.33073805670780839804497214772, 8.452264684880892918012076639256, 8.967544552713325109932539769882, 10.00857783404115280157127025302, 10.92448305492503022204835730889, 12.26050169677894480813023990743

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