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Properties

Label 2-252-28.27-c3-0-28
Degree $2$
Conductor $252$
Sign $0.826 - 0.562i$
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  + (−1.32 + 2.5i)2-s + (−4.5 − 6.61i)4-s + 18.5i·7-s + (22.4 − 2.5i)8-s − 68i·11-s + (−46.3 − 24.5i)14-s + (−23.5 + 59.5i)16-s + (170 + 89.9i)22-s − 40i·23-s + 125·25-s + (122. − 83.3i)28-s + 264.·29-s + (−117. − 137.5i)32-s + 450·37-s + 534. i·43-s + (−449. + 306i)44-s + ⋯
L(s)  = 1  + (−0.467 + 0.883i)2-s + (−0.562 − 0.826i)4-s + 0.999i·7-s + (0.993 − 0.110i)8-s − 1.86i·11-s + (−0.883 − 0.467i)14-s + (−0.367 + 0.930i)16-s + (1.64 + 0.871i)22-s − 0.362i·23-s + 25-s + (0.826 − 0.562i)28-s + 1.69·29-s + (−0.650 − 0.759i)32-s + 1.99·37-s + 1.89i·43-s + (−1.54 + 1.04i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.826 - 0.562i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.826 - 0.562i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.298441626\)
\(L(\frac12)\) \(\approx\) \(1.298441626\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 2.5i)T \)
3 \( 1 \)
7 \( 1 - 18.5iT \)
good5 \( 1 - 125T^{2} \)
11 \( 1 + 68iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 + 40iT - 1.21e4T^{2} \)
29 \( 1 - 264.T + 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 - 450T + 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 - 534. iT - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 - 497.T + 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 + 809. iT - 3.00e5T^{2} \)
71 \( 1 + 688iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 238. iT - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−11.49945048441373363848898223246, −10.66621670801045370582076158131, −9.455891788332461743011031063411, −8.602392341296194828165821508724, −8.033023595290325753363200904508, −6.47332788979309769835405745228, −5.87855304432046433860080313279, −4.71415975279662172551048445016, −2.89941198620534896443329940196, −0.830483988921776054868905206505, 1.01872576455794192301450897955, 2.46833003584558773655603219105, 4.01623261673188693112844234706, 4.82885100754514463026477551141, 6.89859567664378918590195292918, 7.59142916047932473972935945005, 8.783295998870526912867033738287, 9.957882259648347012549286501559, 10.30835267604000425513891664224, 11.46105000508467544847319242445

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