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Properties

Label 2-252-21.17-c11-0-20
Degree $2$
Conductor $252$
Sign $0.939 + 0.341i$
Analytic cond. $193.622$
Root an. cond. $13.9148$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.20e3 + 9.01e3i)5-s + (−3.50e4 + 2.73e4i)7-s + (1.68e5 + 9.70e4i)11-s − 1.37e6i·13-s + (−3.27e6 + 5.67e6i)17-s + (1.51e7 − 8.72e6i)19-s + (−1.35e7 + 7.82e6i)23-s + (−2.98e7 + 5.16e7i)25-s − 1.83e8i·29-s + (9.50e7 + 5.48e7i)31-s + (−4.29e8 − 1.73e8i)35-s + (−2.71e8 − 4.70e8i)37-s + 4.83e8·41-s + 4.53e8·43-s + (−1.27e9 − 2.21e9i)47-s + ⋯
L(s)  = 1  + (0.745 + 1.29i)5-s + (−0.787 + 0.616i)7-s + (0.314 + 0.181i)11-s − 1.03i·13-s + (−0.559 + 0.969i)17-s + (1.40 − 0.808i)19-s + (−0.439 + 0.253i)23-s + (−0.610 + 1.05i)25-s − 1.66i·29-s + (0.596 + 0.344i)31-s + (−1.38 − 0.557i)35-s + (−0.643 − 1.11i)37-s + 0.651·41-s + 0.470·43-s + (−0.813 − 1.40i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.939 + 0.341i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ 0.939 + 0.341i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.982769740\)
\(L(\frac12)\) \(\approx\) \(1.982769740\)
\(L(\frac{13}{2})\) 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 \( 1 + (3.50e4 - 2.73e4i)T \)
good5 \( 1 + (-5.20e3 - 9.01e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-1.68e5 - 9.70e4i)T + (1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 + 1.37e6iT - 1.79e12T^{2} \)
17 \( 1 + (3.27e6 - 5.67e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (-1.51e7 + 8.72e6i)T + (5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (1.35e7 - 7.82e6i)T + (4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 + 1.83e8iT - 1.22e16T^{2} \)
31 \( 1 + (-9.50e7 - 5.48e7i)T + (1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (2.71e8 + 4.70e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 - 4.83e8T + 5.50e17T^{2} \)
43 \( 1 - 4.53e8T + 9.29e17T^{2} \)
47 \( 1 + (1.27e9 + 2.21e9i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (4.42e9 + 2.55e9i)T + (4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (3.14e9 - 5.44e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (-9.92e9 + 5.72e9i)T + (2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (-7.48e9 + 1.29e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 - 8.90e9iT - 2.31e20T^{2} \)
73 \( 1 + (-1.91e10 - 1.10e10i)T + (1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (1.81e10 + 3.13e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 2.60e9T + 1.28e21T^{2} \)
89 \( 1 + (3.01e10 + 5.22e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 1.12e11iT - 7.15e21T^{2} \)
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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

−9.980859723717417992632495603360, −9.401708771941714069559971150730, −8.075983502029910228140326293845, −6.91036708404739139606128928226, −6.19677203746563614551128236080, −5.38716297510867585801148408371, −3.68480458567017559534870088846, −2.81397893633910846342364450428, −2.01713067418303019631608333070, −0.41432705562444703294681192575, 0.885886083966821942716269089017, 1.55655714628462983235462719428, 3.02268748358902223083732164413, 4.26167132826950164549602897687, 5.13687142810632292513937513966, 6.22291689975166383365708079838, 7.14309706324294749712077379867, 8.430294614026701171723365261415, 9.503905942170733020317979978869, 9.668546253836391500980305120984

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