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Properties

Label 2-252-21.2-c8-0-17
Degree $2$
Conductor $252$
Sign $0.943 + 0.332i$
Analytic cond. $102.659$
Root an. cond. $10.1320$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (173. + 99.9i)5-s + (1.61e3 + 1.77e3i)7-s + (2.22e4 − 1.28e4i)11-s + 2.64e4·13-s + (7.42e4 − 4.28e4i)17-s + (−6.94e3 + 1.20e4i)19-s + (−6.09e4 − 3.51e4i)23-s + (−1.75e5 − 3.03e5i)25-s − 4.05e5i·29-s + (−1.30e5 − 2.25e5i)31-s + (1.01e5 + 4.69e5i)35-s + (2.09e5 − 3.62e5i)37-s + 9.27e5i·41-s + 2.87e6·43-s + (−1.51e6 − 8.74e5i)47-s + ⋯
L(s)  = 1  + (0.277 + 0.159i)5-s + (0.671 + 0.740i)7-s + (1.51 − 0.876i)11-s + 0.927·13-s + (0.888 − 0.513i)17-s + (−0.0532 + 0.0922i)19-s + (−0.217 − 0.125i)23-s + (−0.448 − 0.777i)25-s − 0.573i·29-s + (−0.141 − 0.244i)31-s + (0.0676 + 0.312i)35-s + (0.111 − 0.193i)37-s + 0.328i·41-s + 0.842·43-s + (−0.310 − 0.179i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.943 + 0.332i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ 0.943 + 0.332i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.176624433\)
\(L(\frac12)\) \(\approx\) \(3.176624433\)
\(L(5)\) 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 \( 1 + (-1.61e3 - 1.77e3i)T \)
good5 \( 1 + (-173. - 99.9i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-2.22e4 + 1.28e4i)T + (1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 2.64e4T + 8.15e8T^{2} \)
17 \( 1 + (-7.42e4 + 4.28e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (6.94e3 - 1.20e4i)T + (-8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (6.09e4 + 3.51e4i)T + (3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 + 4.05e5iT - 5.00e11T^{2} \)
31 \( 1 + (1.30e5 + 2.25e5i)T + (-4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-2.09e5 + 3.62e5i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 9.27e5iT - 7.98e12T^{2} \)
43 \( 1 - 2.87e6T + 1.16e13T^{2} \)
47 \( 1 + (1.51e6 + 8.74e5i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-3.55e6 + 2.04e6i)T + (3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (7.54e6 - 4.35e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (4.68e6 - 8.12e6i)T + (-9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (1.71e7 + 2.97e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 9.42e6iT - 6.45e14T^{2} \)
73 \( 1 + (1.48e7 + 2.56e7i)T + (-4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (-3.39e6 + 5.88e6i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 6.35e7iT - 2.25e15T^{2} \)
89 \( 1 + (-4.29e7 - 2.47e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 1.06e8T + 7.83e15T^{2} \)
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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

−10.66978893713366733009702031645, −9.432638903267779077199589291358, −8.709592534475473275333410838087, −7.77013887654345474611607845431, −6.29578819877013922882201593357, −5.77098622568773522005374323586, −4.32952544792690144724360419742, −3.20567947589006069904557734338, −1.84047630864178813182504565647, −0.801413196856335762341573907976, 1.12187114824317368001569314183, 1.68383275809998793162452831765, 3.57771279473246089249551668924, 4.36833691006639966347804796460, 5.64920437141819719455826808465, 6.75508737421486550549644904709, 7.68831229181083572851620183942, 8.798771003617813793681650461383, 9.691810649424065587765311988949, 10.67973793015915965593688660868

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