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Properties

Label 2-252-63.25-c1-0-7
Degree $2$
Conductor $252$
Sign $-0.0969 + 0.995i$
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  + (0.570 − 1.63i)3-s + (0.764 − 1.32i)5-s + (−1.91 − 1.82i)7-s + (−2.34 − 1.86i)9-s + (−0.417 − 0.723i)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 + (−4.07 + 2.10i)21-s + (3.07 − 5.32i)23-s + (1.33 + 2.30i)25-s + (−4.39 + 2.77i)27-s + (4.99 − 8.65i)29-s − 3.30·31-s + ⋯
L(s)  = 1  + (0.329 − 0.944i)3-s + (0.341 − 0.592i)5-s + (−0.725 − 0.688i)7-s + (−0.783 − 0.621i)9-s + (−0.125 − 0.218i)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 + (−0.888 + 0.458i)21-s + (0.640 − 1.10i)23-s + (0.266 + 0.460i)25-s + (−0.845 + 0.534i)27-s + (0.927 − 1.60i)29-s − 0.594·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0969 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0969 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.0969 + 0.995i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.0969 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.880966 - 0.970916i\)
\(L(\frac12)\) \(\approx\) \(0.880966 - 0.970916i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.570 + 1.63i)T \)
7 \( 1 + (1.91 + 1.82i)T \)
good5 \( 1 + (-0.764 + 1.32i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.417 + 0.723i)T + (-5.5 + 9.52i)T^{2} \)
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 \( 1 + (-3.07 + 5.32i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.99 + 8.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.30T + 31T^{2} \)
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 \( 1 - 8.46T + 47T^{2} \)
53 \( 1 + (3.99 - 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.130T + 59T^{2} \)
61 \( 1 + 4.76T + 61T^{2} \)
67 \( 1 + 2.24T + 67T^{2} \)
71 \( 1 + 9.39T + 71T^{2} \)
73 \( 1 + (2.25 - 3.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
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

−12.01314329141057827876269646652, −10.93110500971199653888708572371, −9.666351726428150926606385573158, −8.864984695036209799484540268164, −7.84390508959387679713340537353, −6.74518203923294113487237008300, −5.97574289874586862929175447000, −4.32903839748405230085888996345, −2.83420961613768087635905577744, −1.10161820228680079339764307731, 2.65448382001076245664708238447, 3.54185640376500686899125784813, 5.16853409888378652985369723602, 6.04913899756558406661469428812, 7.39282946554522678841609285998, 8.737403478007060848085126398025, 9.408031047378261620488675019008, 10.42479256904794417937465008827, 10.99101910132092855427080880747, 12.33543573239288735432158544036

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