Properties

Label 2-252-7.4-c3-0-7
Degree $2$
Conductor $252$
Sign $-0.269 + 0.963i$
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  + (−0.723 − 1.25i)5-s + (−12.3 + 13.7i)7-s + (23.0 − 39.9i)11-s − 32.2·13-s + (38.8 − 67.3i)17-s + (−6.33 − 10.9i)19-s + (−50.4 − 87.4i)23-s + (61.4 − 106. i)25-s − 213.·29-s + (−21.0 + 36.4i)31-s + (26.1 + 5.56i)35-s + (−155. − 268. i)37-s − 44.0·41-s + 381.·43-s + (−179. − 310. i)47-s + ⋯
L(s)  = 1  + (−0.0646 − 0.112i)5-s + (−0.669 + 0.743i)7-s + (0.632 − 1.09i)11-s − 0.687·13-s + (0.554 − 0.961i)17-s + (−0.0764 − 0.132i)19-s + (−0.457 − 0.792i)23-s + (0.491 − 0.851i)25-s − 1.36·29-s + (−0.121 + 0.211i)31-s + (0.126 + 0.0268i)35-s + (−0.688 − 1.19i)37-s − 0.167·41-s + 1.35·43-s + (−0.555 − 0.962i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.269 + 0.963i$
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: \((2,\ 252,\ (\ :3/2),\ -0.269 + 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.025539713\)
\(L(\frac12)\) \(\approx\) \(1.025539713\)
\(L(\frac{5}{2})\) 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 \( 1 + (12.3 - 13.7i)T \)
good5 \( 1 + (0.723 + 1.25i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-23.0 + 39.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 32.2T + 2.19e3T^{2} \)
17 \( 1 + (-38.8 + 67.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (6.33 + 10.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (50.4 + 87.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 213.T + 2.43e4T^{2} \)
31 \( 1 + (21.0 - 36.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (155. + 268. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 44.0T + 6.89e4T^{2} \)
43 \( 1 - 381.T + 7.95e4T^{2} \)
47 \( 1 + (179. + 310. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-92.4 + 160. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-227. + 393. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (5.92 + 10.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (295. - 511. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 494.T + 3.57e5T^{2} \)
73 \( 1 + (487. - 844. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (149. + 259. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.40e3T + 5.71e5T^{2} \)
89 \( 1 + (347. + 602. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 481.T + 9.12e5T^{2} \)
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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

−11.49345661433319636613613533954, −10.30212530580189652592835295264, −9.276027492595936092764411878352, −8.601550147966004062059628429490, −7.28446770746705004066564893198, −6.19201184052812069689139503759, −5.23278970930816925473995755337, −3.69692071881759325259472680495, −2.46782145476360126172185158771, −0.40455580058650805894299539063, 1.59706414907624448631573874254, 3.38630150627520310377486275048, 4.42307114227122256890539326728, 5.87154697919865649606555660428, 7.05123639630628479690641256614, 7.69466378693065326644719018936, 9.244255383556777798132880917147, 9.912260016402688598032865493337, 10.81316947448976963706839100014, 12.01553689395935616944159556985

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