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Properties

Label 2-252-7.2-c3-0-7
Degree $2$
Conductor $252$
Sign $-0.605 + 0.795i$
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  + (−9.81 + 16.9i)5-s + (3.5 − 18.1i)7-s + (9.81 + 16.9i)11-s − 54·13-s + (−19.6 − 33.9i)17-s + (−1 + 1.73i)19-s + (98.1 − 169. i)23-s + (−130. − 225. i)25-s − 98.1·29-s + (−100.5 − 174. i)31-s + (274. + 237. i)35-s + (101 − 174. i)37-s − 470.·41-s − 244·43-s + (−117. + 203. i)47-s + ⋯
L(s)  = 1  + (−0.877 + 1.51i)5-s + (0.188 − 0.981i)7-s + (0.268 + 0.465i)11-s − 1.15·13-s + (−0.279 − 0.484i)17-s + (−0.0120 + 0.0209i)19-s + (0.889 − 1.54i)23-s + (−1.04 − 1.80i)25-s − 0.628·29-s + (−0.582 − 1.00i)31-s + (1.32 + 1.14i)35-s + (0.448 − 0.777i)37-s − 1.79·41-s − 0.865·43-s + (−0.365 + 0.632i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3415625365\)
\(L(\frac12)\) \(\approx\) \(0.3415625365\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (-3.5 + 18.1i)T \)
good5 \( 1 + (9.81 - 16.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-9.81 - 16.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 54T + 2.19e3T^{2} \)
17 \( 1 + (19.6 + 33.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-98.1 + 169. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 98.1T + 2.43e4T^{2} \)
31 \( 1 + (100.5 + 174. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-101 + 174. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 470.T + 6.89e4T^{2} \)
43 \( 1 + 244T + 7.95e4T^{2} \)
47 \( 1 + (117. - 203. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-323. - 560. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (304. + 526. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-294 + 509. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-151 - 261. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 156.T + 3.57e5T^{2} \)
73 \( 1 + (-223 - 386. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-133.5 + 231. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 725.T + 5.71e5T^{2} \)
89 \( 1 + (58.8 - 101. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 595T + 9.12e5T^{2} \)
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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

−11.14372146288352116683813114079, −10.50389832793200603220406835213, −9.563815516588996025212309989270, −7.981392442041803081205182117016, −7.16724015964520040040326211125, −6.67126008857050997191430789484, −4.75666685706617746851588198888, −3.71370592400233360931577904083, −2.46839515650623840412892435090, −0.13397200527784008886907156109, 1.57516009968359241277080048809, 3.45951102550312350940473824558, 4.86113864003008265101772054209, 5.46537440266212356541014304456, 7.13565736742979967032702202252, 8.306900718158293027444875763632, 8.838589092601051185114130027585, 9.771268620829054013692657113915, 11.41499590115217414121162391682, 11.90062563434127242414135356193

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