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Properties

Label 2-5292-63.25-c1-0-20
Degree $2$
Conductor $5292$
Sign $-0.140 - 0.990i$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 + 2.13i)5-s + (2.32 + 4.02i)11-s + (3.55 + 6.15i)13-s + (2.25 − 3.90i)17-s + (2.16 + 3.74i)19-s + (2.93 − 5.08i)23-s + (−0.527 − 0.912i)25-s + (−3.48 + 6.04i)29-s + 7.38·31-s + (0.363 + 0.629i)37-s + (−0.136 − 0.236i)41-s + (2.41 − 4.18i)43-s + 3.67·47-s + (2.52 − 4.37i)53-s − 11.4·55-s + ⋯
L(s)  = 1  + (−0.550 + 0.952i)5-s + (0.700 + 1.21i)11-s + (0.985 + 1.70i)13-s + (0.547 − 0.948i)17-s + (0.496 + 0.859i)19-s + (0.611 − 1.05i)23-s + (−0.105 − 0.182i)25-s + (−0.647 + 1.12i)29-s + 1.32·31-s + (0.0597 + 0.103i)37-s + (−0.0213 − 0.0369i)41-s + (0.368 − 0.638i)43-s + 0.535·47-s + (0.347 − 0.601i)53-s − 1.54·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.140 - 0.990i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (2125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.140 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.129031048\)
\(L(\frac12)\) \(\approx\) \(2.129031048\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.23 - 2.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.32 - 4.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.55 - 6.15i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.25 + 3.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.16 - 3.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.93 + 5.08i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.48 - 6.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.38T + 31T^{2} \)
37 \( 1 + (-0.363 - 0.629i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.136 + 0.236i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 + 4.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.67T + 47T^{2} \)
53 \( 1 + (-2.52 + 4.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.13T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 1.32T + 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + (2.16 - 3.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 6.43T + 79T^{2} \)
83 \( 1 + (0.742 - 1.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.91 + 8.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.246 - 0.426i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.454590104164669431307535942138, −7.36625783459059910619477426499, −6.97234336510251591798116635696, −6.56700231617525884117403569404, −5.51984334867229347348914727321, −4.51548337672354602661356590376, −3.94704638453354656606083358771, −3.17494736124709141206622677122, −2.16232760142016531584144585903, −1.17990430847891543816193802704, 0.76906473616783038635494682148, 1.10190249055960554256725067079, 2.80035889054993851431617375727, 3.57256893721706707095463701911, 4.14124555687336704352872391466, 5.24376715873258131499273667751, 5.74659492798971202735528442737, 6.38586592247769361931511004132, 7.55254446106186766433487787258, 8.108498166888439352633497035017

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