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Properties

Label 2-252-7.4-c3-0-2
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: $\chi_{252} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.651832220\)
\(L(\frac12)\) \(\approx\) \(1.651832220\)
\(L(\frac{5}{2})\) 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

−11.96469093975945290247866253497, −10.82253232922081207541778576458, −10.06684315399346960573359754696, −9.300437060885156807030174911869, −7.890215065846854013450269710152, −6.83831574672218870126042385341, −5.99021531522195110303543430526, −4.82948380855040286715132314409, −2.85939335964358208518371418678, −2.22291068997121617426531803472, 0.62815749314091174127123133737, 1.95636050566667802057715614801, 3.97363654614933803426528433616, 5.07788522232706180584881873477, 5.91396919868294139377776527580, 7.47634216414866506911116059028, 8.314485441672448491860585599149, 9.535134055442579607726390736885, 10.00233658354703304272884415600, 11.29986129929466061896790374288

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