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Properties

Label 2-252-63.4-c1-0-2
Degree $2$
Conductor $252$
Sign $0.805 - 0.593i$
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  + (1.64 + 0.541i)3-s − 0.763·5-s + (−1.05 + 2.42i)7-s + (2.41 + 1.78i)9-s + 6.03·11-s + (−1.26 + 2.18i)13-s + (−1.25 − 0.413i)15-s + (1.94 − 3.36i)17-s + (−2.13 − 3.69i)19-s + (−3.05 + 3.41i)21-s − 1.46·23-s − 4.41·25-s + (3.00 + 4.23i)27-s + (−3.00 − 5.20i)29-s + (−3.28 − 5.68i)31-s + ⋯
L(s)  = 1  + (0.949 + 0.312i)3-s − 0.341·5-s + (−0.399 + 0.916i)7-s + (0.804 + 0.594i)9-s + 1.81·11-s + (−0.349 + 0.605i)13-s + (−0.324 − 0.106i)15-s + (0.471 − 0.816i)17-s + (−0.489 − 0.848i)19-s + (−0.666 + 0.745i)21-s − 0.305·23-s − 0.883·25-s + (0.578 + 0.815i)27-s + (−0.558 − 0.967i)29-s + (−0.589 − 1.02i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.805 - 0.593i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.805 - 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53614 + 0.504792i\)
\(L(\frac12)\) \(\approx\) \(1.53614 + 0.504792i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 0.541i)T \)
7 \( 1 + (1.05 - 2.42i)T \)
good5 \( 1 + 0.763T + 5T^{2} \)
11 \( 1 - 6.03T + 11T^{2} \)
13 \( 1 + (1.26 - 2.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.94 + 3.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.13 + 3.69i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.46T + 23T^{2} \)
29 \( 1 + (3.00 + 5.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.28 + 5.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.82 - 8.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.24 - 3.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.13 + 3.69i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.38 + 5.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.265 - 0.459i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.59 + 9.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.19 - 7.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.961 - 1.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.90T + 71T^{2} \)
73 \( 1 + (2.13 - 3.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.70 + 6.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.05 + 13.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.76 - 3.05i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.33 - 4.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.93703345502308778811660634889, −11.49508832995927736254921141351, −9.721001469722567546765989342442, −9.368768072403428534560157064648, −8.449357075172330766250139667434, −7.27456667500121985233722107685, −6.20082963820742460892593506847, −4.57011213432355629307231836594, −3.54911138506106296682700589274, −2.14997621336637304319313741056, 1.50549989478228056310254925555, 3.52150028846866489960200825531, 4.06444593452229331725387629766, 6.12907080022270689750308891229, 7.15355090863509237868229376238, 7.956115501523046891056516968977, 9.048180643583693102195157114581, 9.884104875075093367595322797441, 10.89937188407847168516677334468, 12.30412194566115442642173937993

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