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Properties

Label 2-252-9.7-c3-0-7
Degree $2$
Conductor $252$
Sign $0.335 + 0.942i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.01 − 1.36i)3-s + (−0.333 − 0.577i)5-s + (−3.5 + 6.06i)7-s + (23.2 + 13.6i)9-s + (−6.42 + 11.1i)11-s + (−2.93 − 5.08i)13-s + (0.883 + 3.34i)15-s + 10.4·17-s + 25.8·19-s + (25.8 − 25.6i)21-s + (−32.0 − 55.4i)23-s + (62.2 − 107. i)25-s + (−98.0 − 100. i)27-s + (81.4 − 141. i)29-s + (−59.6 − 103. i)31-s + ⋯
L(s)  = 1  + (−0.964 − 0.262i)3-s + (−0.0297 − 0.0516i)5-s + (−0.188 + 0.327i)7-s + (0.862 + 0.506i)9-s + (−0.176 + 0.304i)11-s + (−0.0625 − 0.108i)13-s + (0.0152 + 0.0576i)15-s + 0.149·17-s + 0.312·19-s + (0.268 − 0.266i)21-s + (−0.290 − 0.502i)23-s + (0.498 − 0.862i)25-s + (−0.699 − 0.714i)27-s + (0.521 − 0.903i)29-s + (−0.345 − 0.598i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.335 + 0.942i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.335 + 0.942i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.009312415\)
\(L(\frac12)\) \(\approx\) \(1.009312415\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.01 + 1.36i)T \)
7 \( 1 + (3.5 - 6.06i)T \)
good5 \( 1 + (0.333 + 0.577i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (6.42 - 11.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (2.93 + 5.08i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 10.4T + 4.91e3T^{2} \)
19 \( 1 - 25.8T + 6.85e3T^{2} \)
23 \( 1 + (32.0 + 55.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-81.4 + 141. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (59.6 + 103. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 292.T + 5.06e4T^{2} \)
41 \( 1 + (196. + 340. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-70.7 + 122. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-99.6 + 172. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 375.T + 1.48e5T^{2} \)
59 \( 1 + (226. + 392. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-126. + 219. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-432. - 749. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 712.T + 3.57e5T^{2} \)
73 \( 1 - 527.T + 3.89e5T^{2} \)
79 \( 1 + (142. - 246. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (29.5 - 51.1i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 189.T + 7.04e5T^{2} \)
97 \( 1 + (-143. + 249. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.55558236868517446023444362478, −10.48724513421404392816209822645, −9.739326321332954385950808346487, −8.385778873846985684720127680671, −7.29633021495227651287652890845, −6.28406641406298804026899370609, −5.34241432152687044541664626800, −4.18912257808740991131058450749, −2.31647358522376937413641457480, −0.54261886397059207222944854151, 1.13773816709273365534160212372, 3.29249893114355676117769142628, 4.60899917743467570255280616472, 5.64337342443943323890571811953, 6.68951823238427031624281302609, 7.66034141742898052835596211729, 9.095600263715821394729875418160, 10.04028004258153826292762816840, 10.89119023287338090274284417835, 11.65101984183247879810359044722

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