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Properties

Label 2-252-63.59-c3-0-18
Degree $2$
Conductor $252$
Sign $0.931 + 0.363i$
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  + (5.16 + 0.525i)3-s + 10.4·5-s + (15.5 − 10.0i)7-s + (26.4 + 5.43i)9-s − 53.8i·11-s + (−25.4 − 14.6i)13-s + (53.7 + 5.46i)15-s + (−39.8 + 69.0i)17-s + (25.7 − 14.8i)19-s + (85.5 − 43.9i)21-s − 1.07i·23-s − 16.7·25-s + (133. + 41.9i)27-s + (91.0 − 52.5i)29-s + (−109. + 63.0i)31-s + ⋯
L(s)  = 1  + (0.994 + 0.101i)3-s + 0.930·5-s + (0.838 − 0.544i)7-s + (0.979 + 0.201i)9-s − 1.47i·11-s + (−0.542 − 0.312i)13-s + (0.925 + 0.0941i)15-s + (−0.568 + 0.984i)17-s + (0.310 − 0.179i)19-s + (0.889 − 0.457i)21-s − 0.00971i·23-s − 0.133·25-s + (0.954 + 0.299i)27-s + (0.582 − 0.336i)29-s + (−0.632 + 0.365i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.931 + 0.363i$
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: \((2,\ 252,\ (\ :3/2),\ 0.931 + 0.363i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.175969129\)
\(L(\frac12)\) \(\approx\) \(3.175969129\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.16 - 0.525i)T \)
7 \( 1 + (-15.5 + 10.0i)T \)
good5 \( 1 - 10.4T + 125T^{2} \)
11 \( 1 + 53.8iT - 1.33e3T^{2} \)
13 \( 1 + (25.4 + 14.6i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (39.8 - 69.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-25.7 + 14.8i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 1.07iT - 1.21e4T^{2} \)
29 \( 1 + (-91.0 + 52.5i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (109. - 63.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-42.3 - 73.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (121. - 210. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-201. - 349. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-59.8 + 103. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-119. - 68.7i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (347. + 601. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-322. - 186. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-336. - 583. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 139. iT - 3.57e5T^{2} \)
73 \( 1 + (-215. - 124. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (574. - 995. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-263. - 455. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (416. + 720. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.09e3 - 633. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.32353456077116782251805386179, −10.47690035006704738895091718323, −9.599244966600320015172036272844, −8.533972583718766039193136926364, −7.896053504378014391849145870095, −6.56007895322790962034751289903, −5.31149206484882065323975795358, −4.00136566327443702338367088126, −2.65910538647462749526218736737, −1.33238215984939064013253197972, 1.80128509382907656894262245467, 2.47554546628116776158312278431, 4.34467842919602475559157608316, 5.34428168677100683062023547536, 6.91987811966177382700625165824, 7.67957971750359432917305832198, 8.991657839843176483036376032719, 9.501374144772350805122307203856, 10.43725813700361238461926139299, 11.87640586159523460847016007602

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