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Properties

Label 2-252-63.59-c3-0-6
Degree $2$
Conductor $252$
Sign $0.715 - 0.699i$
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  + (0.222 − 5.19i)3-s − 3.95·5-s + (0.649 + 18.5i)7-s + (−26.9 − 2.31i)9-s + 1.45i·11-s + (54.0 + 31.2i)13-s + (−0.882 + 20.5i)15-s + (−65.5 + 113. i)17-s + (32.4 − 18.7i)19-s + (96.2 + 0.756i)21-s + 35.2i·23-s − 109.·25-s + (−18.0 + 139. i)27-s + (252. − 145. i)29-s + (167. − 96.4i)31-s + ⋯
L(s)  = 1  + (0.0429 − 0.999i)3-s − 0.354·5-s + (0.0350 + 0.999i)7-s + (−0.996 − 0.0857i)9-s + 0.0398i·11-s + (1.15 + 0.666i)13-s + (−0.0151 + 0.353i)15-s + (−0.935 + 1.62i)17-s + (0.392 − 0.226i)19-s + (0.999 + 0.00785i)21-s + 0.319i·23-s − 0.874·25-s + (−0.128 + 0.991i)27-s + (1.61 − 0.932i)29-s + (0.967 − 0.558i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.715 - 0.699i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.715 - 0.699i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.378502299\)
\(L(\frac12)\) \(\approx\) \(1.378502299\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.222 + 5.19i)T \)
7 \( 1 + (-0.649 - 18.5i)T \)
good5 \( 1 + 3.95T + 125T^{2} \)
11 \( 1 - 1.45iT - 1.33e3T^{2} \)
13 \( 1 + (-54.0 - 31.2i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (65.5 - 113. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-32.4 + 18.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 - 35.2iT - 1.21e4T^{2} \)
29 \( 1 + (-252. + 145. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-167. + 96.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-204. - 354. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-66.7 + 115. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-215. - 372. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (175. - 304. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (422. + 244. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-44.3 - 76.8i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (540. + 311. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-294. - 510. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 1.01e3iT - 3.57e5T^{2} \)
73 \( 1 + (-559. - 323. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (214. - 371. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (593. + 1.02e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (347. + 602. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (263. - 152. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.64134955644820735835992829287, −11.23799646677149988336689718294, −9.635460001487759209576082111948, −8.431539535989268238822153577715, −8.083183669056656735044772504528, −6.44072366388036053208881347941, −6.05629551074950490383014113365, −4.32283118584570627468470079821, −2.74281010916170562122025061023, −1.42835754467075723068698747999, 0.58401126144021042772885775709, 3.00173718911385159002829255672, 4.07286018138204220262699076672, 5.02005295042790393254995885024, 6.39504210316259958110001244730, 7.65605564275549431666946316186, 8.654729302817963280331656938997, 9.642045356458381534395119441607, 10.66581076369531064780054273083, 11.14542457880179236477012730212

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