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Properties

Label 2-252-9.4-c1-0-3
Degree $2$
Conductor $252$
Sign $0.787 + 0.616i$
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  + (−1.09 + 1.34i)3-s + (1.97 − 3.41i)5-s + (−0.5 − 0.866i)7-s + (−0.619 − 2.93i)9-s + (−0.471 − 0.816i)11-s + (0.5 − 0.866i)13-s + (2.44 + 6.37i)15-s + 5.60·17-s + 1.28·19-s + (1.71 + 0.272i)21-s + (2.33 − 4.03i)23-s + (−5.27 − 9.13i)25-s + (4.62 + 2.36i)27-s + (3.83 + 6.63i)29-s + (−3.91 + 6.77i)31-s + ⋯
L(s)  = 1  + (−0.629 + 0.776i)3-s + (0.881 − 1.52i)5-s + (−0.188 − 0.327i)7-s + (−0.206 − 0.978i)9-s + (−0.142 − 0.246i)11-s + (0.138 − 0.240i)13-s + (0.630 + 1.64i)15-s + 1.35·17-s + 0.294·19-s + (0.373 + 0.0593i)21-s + (0.485 − 0.841i)23-s + (−1.05 − 1.82i)25-s + (0.890 + 0.455i)27-s + (0.711 + 1.23i)29-s + (−0.703 + 1.21i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07615 - 0.371405i\)
\(L(\frac12)\) \(\approx\) \(1.07615 - 0.371405i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 - 1.34i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.97 + 3.41i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.471 + 0.816i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.60T + 17T^{2} \)
19 \( 1 - 1.28T + 19T^{2} \)
23 \( 1 + (-2.33 + 4.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.83 - 6.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.91 - 6.77i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.82T + 37T^{2} \)
41 \( 1 + (0.471 - 0.816i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.63 + 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.64 - 4.57i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.22T + 53T^{2} \)
59 \( 1 + (-4.77 + 8.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.27 - 9.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.858 + 1.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.54T + 71T^{2} \)
73 \( 1 - 2.71T + 73T^{2} \)
79 \( 1 + (-8.18 - 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.198 - 0.343i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.50T + 89T^{2} \)
97 \( 1 + (-6.77 - 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.25933018310293188268160135043, −10.73936832974154510879324174721, −10.06593381780521181832808361405, −9.149363611572014547056552446971, −8.399255616344608328370134444199, −6.68582092825434791126346096798, −5.37604648533010097552585100069, −5.04327689127611963480938657351, −3.50510187568248245447930128917, −1.08974351314004192448498425590, 1.94399917305330551439172466061, 3.19786250450417210589000744941, 5.38033739547577080378556808304, 6.15693197229532623232883241238, 7.02106708846275704244825594669, 7.88068513161036950406299520486, 9.584589294127870276422824768209, 10.29677333054951841987287979956, 11.26207525869151971052197314641, 12.00175167378172209713509023752

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