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Properties

Label 2-252-63.5-c1-0-2
Degree $2$
Conductor $252$
Sign $0.993 - 0.114i$
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  + (−1.63 + 0.576i)3-s + (0.0382 + 0.0661i)5-s + (0.232 − 2.63i)7-s + (2.33 − 1.88i)9-s + (4.66 + 2.69i)11-s + (4.60 + 2.65i)13-s + (−0.100 − 0.0860i)15-s + (1.89 + 3.27i)17-s + (−4.33 − 2.50i)19-s + (1.13 + 4.43i)21-s + (−2.02 + 1.16i)23-s + (2.49 − 4.32i)25-s + (−2.73 + 4.42i)27-s + (8.84 − 5.10i)29-s − 5.74i·31-s + ⋯
L(s)  = 1  + (−0.943 + 0.332i)3-s + (0.0170 + 0.0295i)5-s + (0.0880 − 0.996i)7-s + (0.778 − 0.627i)9-s + (1.40 + 0.811i)11-s + (1.27 + 0.737i)13-s + (−0.0259 − 0.0222i)15-s + (0.458 + 0.794i)17-s + (−0.995 − 0.574i)19-s + (0.248 + 0.968i)21-s + (−0.422 + 0.243i)23-s + (0.499 − 0.865i)25-s + (−0.525 + 0.850i)27-s + (1.64 − 0.948i)29-s − 1.03i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.993 - 0.114i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.993 - 0.114i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05079 + 0.0601936i\)
\(L(\frac12)\) \(\approx\) \(1.05079 + 0.0601936i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.63 - 0.576i)T \)
7 \( 1 + (-0.232 + 2.63i)T \)
good5 \( 1 + (-0.0382 - 0.0661i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.66 - 2.69i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.60 - 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.89 - 3.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.33 + 2.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.02 - 1.16i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.84 + 5.10i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.74iT - 31T^{2} \)
37 \( 1 + (-0.354 + 0.613i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.29 - 5.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.716 - 1.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.92T + 47T^{2} \)
53 \( 1 + (10.4 - 6.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.579T + 59T^{2} \)
61 \( 1 + 2.77iT - 61T^{2} \)
67 \( 1 - 5.27T + 67T^{2} \)
71 \( 1 - 3.32iT - 71T^{2} \)
73 \( 1 + (6.17 - 3.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.938T + 79T^{2} \)
83 \( 1 + (-6.49 - 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.51 + 2.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.18 - 3.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.88198484913830228379439674853, −11.13019144372785757074534813101, −10.27162147213595330033690806648, −9.424571668623167847875911642465, −8.142704625257016293110731627989, −6.56815087334364038634763053072, −6.38454324505033856992922275314, −4.44929434355521404436287221871, −4.00803611995973072516884756960, −1.32458985320243722622340476834, 1.34057058194336756812317129665, 3.40282029556740105020468740761, 5.02168646501162384818214367003, 6.03948830214721789266620417112, 6.67546923520943351006631582963, 8.261753602747367105924577778682, 8.976986351093143766178255674414, 10.38052821592899126044468341660, 11.20661577471051836184813771718, 12.02848912581263324776142719832

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