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Properties

Label 2-252-63.59-c1-0-7
Degree $2$
Conductor $252$
Sign $-0.855 + 0.517i$
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  + (−1.31 − 1.12i)3-s + 0.0764·5-s + (−2.39 − 1.11i)7-s + (0.462 + 2.96i)9-s − 5.38i·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 + (1.89 + 4.17i)21-s − 2.33i·23-s − 4.99·25-s + (2.73 − 4.42i)27-s + (8.84 − 5.10i)29-s + (4.97 − 2.87i)31-s + ⋯
L(s)  = 1  + (−0.759 − 0.650i)3-s + 0.0341·5-s + (−0.906 − 0.421i)7-s + (0.154 + 0.988i)9-s − 1.62i·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 + (0.414 + 0.910i)21-s − 0.487i·23-s − 0.998·25-s + (0.525 − 0.850i)27-s + (1.64 − 0.948i)29-s + (0.893 − 0.516i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.855 + 0.517i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138346 - 0.495778i\)
\(L(\frac12)\) \(\approx\) \(0.138346 - 0.495778i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 + 1.12i)T \)
7 \( 1 + (2.39 + 1.11i)T \)
good5 \( 1 - 0.0764T + 5T^{2} \)
11 \( 1 + 5.38iT - 11T^{2} \)
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 \( 1 + 2.33iT - 23T^{2} \)
29 \( 1 + (-8.84 + 5.10i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.97 + 2.87i)T + (15.5 - 26.8i)T^{2} \)
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 \( 1 + (1.46 - 2.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.4 - 6.05i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.289 + 0.502i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.40 + 1.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.63 + 4.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.32iT - 71T^{2} \)
73 \( 1 + (6.17 + 3.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.469 - 0.812i)T + (-39.5 - 68.4i)T^{2} \)
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

−11.78347174040581891376066120710, −10.60261954908217741847405440726, −10.11554461351832673784461957957, −8.525545833007728362026277731440, −7.64897878740171790041946627122, −6.34625542084578838390532904449, −5.89164009639523865953172338048, −4.29433287546553933107946943958, −2.64944811537694575535911338650, −0.42177171080692285946820330742, 2.52941968447915270475110343846, 4.32095949020050272108745668061, 5.06070075201662529013806869531, 6.53630834238415151361095593466, 7.11086147521085228627278565308, 8.956454615761057903049745994620, 9.757905298880831455306972493151, 10.26475783234982735497698624466, 11.72687817032703678467284988788, 12.18213482296241314827521820049

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