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Properties

Label 2-252-7.2-c1-0-2
Degree $2$
Conductor $252$
Sign $0.386 + 0.922i$
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  + (1.5 − 2.59i)5-s + (−2 − 1.73i)7-s + (−1.5 − 2.59i)11-s + 2·13-s + (1.5 + 2.59i)17-s + (0.5 − 0.866i)19-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s + 6·29-s + (3.5 + 6.06i)31-s + (−7.5 + 2.59i)35-s + (0.5 − 0.866i)37-s − 6·41-s − 4·43-s + (−4.5 + 7.79i)47-s + ⋯
L(s)  = 1  + (0.670 − 1.16i)5-s + (−0.755 − 0.654i)7-s + (−0.452 − 0.783i)11-s + 0.554·13-s + (0.363 + 0.630i)17-s + (0.114 − 0.198i)19-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s + 1.11·29-s + (0.628 + 1.08i)31-s + (−1.26 + 0.439i)35-s + (0.0821 − 0.142i)37-s − 0.937·41-s − 0.609·43-s + (−0.656 + 1.13i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04817 - 0.697228i\)
\(L(\frac12)\) \(\approx\) \(1.04817 - 0.697228i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−12.05721099174022547234879819740, −10.70185343984639502147945569607, −10.00067533311313666691979033844, −8.896270539611800093161382839532, −8.211079863766860731141599490917, −6.69695841326881302583476853477, −5.74162173220900054190464728814, −4.60851187715082967625889052792, −3.16409515102708516884592284828, −1.09900860833751981997528731422, 2.32080880964095605130769555739, 3.35586292307381058452228809307, 5.19497382469695529042734941497, 6.30770230010870993556249534334, 7.02057016779190290075933797540, 8.337764788676738567030829354261, 9.749015658413658137623813785412, 10.03010945048315609642531373718, 11.24913716824266258475995230084, 12.18603632272705957033828963018

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