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Properties

Label 2-252-7.2-c7-0-19
Degree $2$
Conductor $252$
Sign $-0.742 + 0.669i$
Analytic cond. $78.7210$
Root an. cond. $8.87248$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−173. + 299. i)5-s + (334. − 843. i)7-s + (−437. − 758. i)11-s − 507.·13-s + (9.80e3 + 1.69e4i)17-s + (1.77e4 − 3.07e4i)19-s + (−4.87e4 + 8.44e4i)23-s + (−2.08e4 − 3.61e4i)25-s − 5.66e4·29-s + (2.63e4 + 4.56e4i)31-s + (1.95e5 + 2.46e5i)35-s + (−1.71e5 + 2.97e5i)37-s − 2.79e5·41-s + 7.37e4·43-s + (5.14e5 − 8.91e5i)47-s + ⋯
L(s)  = 1  + (−0.619 + 1.07i)5-s + (0.368 − 0.929i)7-s + (−0.0991 − 0.171i)11-s − 0.0640·13-s + (0.483 + 0.838i)17-s + (0.594 − 1.02i)19-s + (−0.835 + 1.44i)23-s + (−0.267 − 0.463i)25-s − 0.431·29-s + (0.158 + 0.274i)31-s + (0.769 + 0.970i)35-s + (−0.558 + 0.966i)37-s − 0.633·41-s + 0.141·43-s + (0.723 − 1.25i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.742 + 0.669i$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ -0.742 + 0.669i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.3170128839\)
\(L(\frac12)\) \(\approx\) \(0.3170128839\)
\(L(\frac{9}{2})\) 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 \( 1 + (-334. + 843. i)T \)
good5 \( 1 + (173. - 299. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (437. + 758. i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + 507.T + 6.27e7T^{2} \)
17 \( 1 + (-9.80e3 - 1.69e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-1.77e4 + 3.07e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (4.87e4 - 8.44e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + 5.66e4T + 1.72e10T^{2} \)
31 \( 1 + (-2.63e4 - 4.56e4i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (1.71e5 - 2.97e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + 2.79e5T + 1.94e11T^{2} \)
43 \( 1 - 7.37e4T + 2.71e11T^{2} \)
47 \( 1 + (-5.14e5 + 8.91e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (1.86e5 + 3.23e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (5.20e5 + 9.01e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-1.61e5 + 2.79e5i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (1.50e6 + 2.60e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 2.97e6T + 9.09e12T^{2} \)
73 \( 1 + (1.26e6 + 2.19e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-9.36e5 + 1.62e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + 3.44e6T + 2.71e13T^{2} \)
89 \( 1 + (5.03e6 - 8.71e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.52e7T + 8.07e13T^{2} \)
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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

−10.58356121693807119256740189972, −9.679954961890300835068462501585, −8.209017747820723942122358064467, −7.43279171260964276469615026704, −6.68057142722241790176330562035, −5.28840830105591467061769513879, −3.91209330912792132534865203931, −3.16118571224497916320979507891, −1.55093840107216416976266542117, −0.07796492368229624761206507996, 1.19370101748091007767074527603, 2.53893700410287288746494213731, 4.05388599431701583853179680927, 5.04235193916409416342815077282, 5.92231538218799042843878169253, 7.49836745461527713403653919824, 8.309582241130931347954621769001, 9.073554508409966954467580873758, 10.09964219390395515598912242011, 11.41066554561181924580494550224

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