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Properties

Label 2-252-63.25-c3-0-20
Degree $2$
Conductor $252$
Sign $0.140 + 0.990i$
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  + (5.17 + 0.446i)3-s + (3.12 − 5.42i)5-s + (−9.69 − 15.7i)7-s + (26.6 + 4.62i)9-s + (−28.4 − 49.3i)11-s + (7.74 + 13.4i)13-s + (18.6 − 26.6i)15-s + (10.1 − 17.6i)17-s + (−67.1 − 116. i)19-s + (−43.1 − 86.0i)21-s + (−8.28 + 14.3i)23-s + (42.9 + 74.3i)25-s + (135. + 35.8i)27-s + (2.32 − 4.03i)29-s + 135.·31-s + ⋯
L(s)  = 1  + (0.996 + 0.0859i)3-s + (0.279 − 0.484i)5-s + (−0.523 − 0.851i)7-s + (0.985 + 0.171i)9-s + (−0.781 − 1.35i)11-s + (0.165 + 0.286i)13-s + (0.320 − 0.458i)15-s + (0.145 − 0.251i)17-s + (−0.811 − 1.40i)19-s + (−0.448 − 0.893i)21-s + (−0.0751 + 0.130i)23-s + (0.343 + 0.594i)25-s + (0.966 + 0.255i)27-s + (0.0149 − 0.0258i)29-s + 0.786·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.140 + 0.990i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.140 + 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.252009194\)
\(L(\frac12)\) \(\approx\) \(2.252009194\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.17 - 0.446i)T \)
7 \( 1 + (9.69 + 15.7i)T \)
good5 \( 1 + (-3.12 + 5.42i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (28.4 + 49.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-7.74 - 13.4i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-10.1 + 17.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (67.1 + 116. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (8.28 - 14.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-2.32 + 4.03i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 135.T + 2.97e4T^{2} \)
37 \( 1 + (97.4 + 168. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-120. - 208. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (18.6 - 32.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 360.T + 1.03e5T^{2} \)
53 \( 1 + (-321. + 556. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 - 262.T + 2.05e5T^{2} \)
61 \( 1 + 371.T + 2.26e5T^{2} \)
67 \( 1 + 307.T + 3.00e5T^{2} \)
71 \( 1 - 323.T + 3.57e5T^{2} \)
73 \( 1 + (-331. + 574. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + 314.T + 4.93e5T^{2} \)
83 \( 1 + (115. - 199. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (-695. - 1.20e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (817. - 1.41e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.15375712880818153405770031930, −10.36113782951960977196736333237, −9.281934057298649621380954281405, −8.597056941723344690090899360156, −7.55927963702227503431864309873, −6.47760225083427211653604183991, −4.99279897579206489479812658577, −3.74714202471203066174519986688, −2.62194624638825717160865605108, −0.789258292086531137192558363393, 2.00328734073199996022804043105, 2.88953603722332317521233825997, 4.29774315641661292916746365317, 5.83396020482636125053090159753, 6.93912477268054391727184127253, 7.991512281398115341637905536477, 8.863783600328330895996076332913, 10.05547345233424677770411245592, 10.36082168305673651172675008167, 12.28416016127463178030194692497

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