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Properties

Label 2-252-63.41-c1-0-2
Degree $2$
Conductor $252$
Sign $0.763 - 0.646i$
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  + (−1.15 − 1.28i)3-s + (−1.21 + 2.10i)5-s + (1.57 + 2.12i)7-s + (−0.324 + 2.98i)9-s + (2.09 − 1.21i)11-s + (4.73 + 2.73i)13-s + (4.10 − 0.866i)15-s − 2.58·17-s − 0.402i·19-s + (0.924 − 4.48i)21-s + (3.06 + 1.77i)23-s + (−0.440 − 0.762i)25-s + (4.21 − 3.03i)27-s + (−6.31 + 3.64i)29-s + (3.63 + 2.10i)31-s + ⋯
L(s)  = 1  + (−0.667 − 0.744i)3-s + (−0.542 + 0.939i)5-s + (0.594 + 0.804i)7-s + (−0.108 + 0.994i)9-s + (0.632 − 0.365i)11-s + (1.31 + 0.758i)13-s + (1.06 − 0.223i)15-s − 0.625·17-s − 0.0923i·19-s + (0.201 − 0.979i)21-s + (0.639 + 0.369i)23-s + (−0.0880 − 0.152i)25-s + (0.812 − 0.583i)27-s + (−1.17 + 0.677i)29-s + (0.653 + 0.377i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.763 - 0.646i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.763 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923356 + 0.338481i\)
\(L(\frac12)\) \(\approx\) \(0.923356 + 0.338481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.15 + 1.28i)T \)
7 \( 1 + (-1.57 - 2.12i)T \)
good5 \( 1 + (1.21 - 2.10i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.09 + 1.21i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.73 - 2.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.58T + 17T^{2} \)
19 \( 1 + 0.402iT - 19T^{2} \)
23 \( 1 + (-3.06 - 1.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.31 - 3.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.63 - 2.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.19T + 37T^{2} \)
41 \( 1 + (4.03 - 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.22 + 7.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.25 - 3.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 14.0iT - 53T^{2} \)
59 \( 1 + (-0.0779 + 0.134i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.2 + 5.90i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.53 + 4.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.73iT - 71T^{2} \)
73 \( 1 + 8.80iT - 73T^{2} \)
79 \( 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.50 + 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (4.97 - 2.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.77207776431115081684395754481, −11.37141402889258347850421779547, −10.76017432117428765245683751199, −9.041493986480807505891832986037, −8.185203671874267728023791390957, −6.94691409804770826800962362454, −6.34082428490628598907242918099, −5.09309324974378297326526163926, −3.48145167061054345163772896916, −1.76591330468639726777043987032, 0.959044660620802922282608852693, 3.82061858242482176445938151836, 4.47914604216709044545090821699, 5.60165525701273654008358772369, 6.87589232665720403790680920074, 8.231060431403586844820890227699, 8.991419568473685029536923335106, 10.19557441868785969948738615107, 11.06278794489138264581080098878, 11.72591267657031377230623316873

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