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Properties

Label 2-252-252.115-c1-0-12
Degree $2$
Conductor $252$
Sign $0.999 - 0.00335i$
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  + (−1.15 + 0.809i)2-s + (−0.630 − 1.61i)3-s + (0.688 − 1.87i)4-s + (2.22 + 1.28i)5-s + (2.03 + 1.36i)6-s + (−2.01 + 1.71i)7-s + (0.722 + 2.73i)8-s + (−2.20 + 2.03i)9-s + (−3.61 + 0.312i)10-s + (4.82 − 2.78i)11-s + (−3.46 + 0.0723i)12-s + (3.36 − 1.94i)13-s + (0.947 − 3.61i)14-s + (0.670 − 4.39i)15-s + (−3.05 − 2.58i)16-s + (2.65 + 1.53i)17-s + ⋯
L(s)  = 1  + (−0.819 + 0.572i)2-s + (−0.363 − 0.931i)3-s + (0.344 − 0.938i)4-s + (0.994 + 0.574i)5-s + (0.831 + 0.555i)6-s + (−0.761 + 0.648i)7-s + (0.255 + 0.966i)8-s + (−0.735 + 0.677i)9-s + (−1.14 + 0.0987i)10-s + (1.45 − 0.839i)11-s + (−0.999 + 0.0208i)12-s + (0.933 − 0.538i)13-s + (0.253 − 0.967i)14-s + (0.173 − 1.13i)15-s + (−0.762 − 0.646i)16-s + (0.642 + 0.371i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.999 - 0.00335i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.999 - 0.00335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.903469 + 0.00151575i\)
\(L(\frac12)\) \(\approx\) \(0.903469 + 0.00151575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.809i)T \)
3 \( 1 + (0.630 + 1.61i)T \)
7 \( 1 + (2.01 - 1.71i)T \)
good5 \( 1 + (-2.22 - 1.28i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.82 + 2.78i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.36 + 1.94i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.65 - 1.53i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.51 + 2.62i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.90 - 2.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.91 + 5.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.712T + 31T^{2} \)
37 \( 1 + (-3.16 - 5.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.66 - 3.27i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.877 + 0.506i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.374T + 47T^{2} \)
53 \( 1 + (2.00 - 3.47i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.31T + 59T^{2} \)
61 \( 1 + 9.20iT - 61T^{2} \)
67 \( 1 - 6.34iT - 67T^{2} \)
71 \( 1 - 12.0iT - 71T^{2} \)
73 \( 1 + (13.8 + 8.00i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.94iT - 79T^{2} \)
83 \( 1 + (4.89 - 8.48i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.02 + 1.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.10 + 1.79i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.82504459768746139358597312434, −11.04657915814398562481010434450, −9.993769518050808660193325863110, −9.027676345832634663238592561057, −8.208857087640005371315055563719, −6.71602531132850991053583122441, −6.27163578932212583518346900305, −5.62635000440252796459083316304, −2.88816144963491112321236245973, −1.30102616047708549839485146269, 1.37402958264485563541577158182, 3.43677615936800000692244037261, 4.44564140287428671209195400816, 6.09926143670987294667997395537, 6.98693107968981100253141463358, 8.822466360241001046620078342803, 9.306010219958673229686341721925, 10.02084771210964789092836421120, 10.79156501302366092326456052995, 11.90710891150831913937782812276

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