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Properties

Label 2-294-7.2-c7-0-27
Degree $2$
Conductor $294$
Sign $0.605 - 0.795i$
Analytic cond. $91.8411$
Root an. cond. $9.58338$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 6.92i)2-s + (−13.5 − 23.3i)3-s + (−31.9 − 55.4i)4-s + (−235 + 407. i)5-s + 216·6-s + 511.·8-s + (−364.5 + 631. i)9-s + (−1.87e3 − 3.25e3i)10-s + (3.63e3 + 6.29e3i)11-s + (−864. + 1.49e3i)12-s + 1.13e4·13-s + 1.26e4·15-s + (−2.04e3 + 3.54e3i)16-s + (−1.04e4 − 1.80e4i)17-s + (−2.91e3 − 5.05e3i)18-s + (6.84e3 − 1.18e4i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.840 + 1.45i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.594 − 1.02i)10-s + (0.823 + 1.42i)11-s + (−0.144 + 0.249i)12-s + 1.43·13-s + 0.970·15-s + (−0.125 + 0.216i)16-s + (−0.514 − 0.891i)17-s + (−0.117 − 0.204i)18-s + (0.228 − 0.396i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(91.8411\)
Root analytic conductor: \(9.58338\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :7/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.364934233\)
\(L(\frac12)\) \(\approx\) \(1.364934233\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 - 6.92i)T \)
3 \( 1 + (13.5 + 23.3i)T \)
7 \( 1 \)
good5 \( 1 + (235 - 407. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
11 \( 1 + (-3.63e3 - 6.29e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 - 1.13e4T + 6.27e7T^{2} \)
17 \( 1 + (1.04e4 + 1.80e4i)T + (-2.05e8 + 3.55e8i)T^{2} \)
19 \( 1 + (-6.84e3 + 1.18e4i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (-2.33e4 + 4.04e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 - 2.77e3T + 1.72e10T^{2} \)
31 \( 1 + (-5.61e3 - 9.72e3i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (-1.19e5 + 2.07e5i)T + (-4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 - 5.29e4T + 1.94e11T^{2} \)
43 \( 1 + 8.74e5T + 2.71e11T^{2} \)
47 \( 1 + (-2.25e5 + 3.90e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 + (-9.71e5 - 1.68e6i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (7.81e5 + 1.35e6i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-1.59e6 + 2.75e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (1.06e6 + 1.84e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 5.69e6T + 9.09e12T^{2} \)
73 \( 1 + (-1.12e6 - 1.95e6i)T + (-5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (6.64e4 - 1.15e5i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 - 6.95e6T + 2.71e13T^{2} \)
89 \( 1 + (-5.31e6 + 9.20e6i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 - 2.40e6T + 8.07e13T^{2} \)
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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

−10.86006216706538622409188130178, −9.727562745139641598057234299112, −8.598333005048052201768003557655, −7.47500009446721796015196845769, −6.87135634342503276146475293179, −6.31425245324834374931841370402, −4.70136688283467577368310452884, −3.52466870744652224539634074109, −2.10390646114507674990508447111, −0.61270414574465112540363598177, 0.72262259556792063577564694109, 1.33770567363112869478181544509, 3.56318226180308977580435819603, 3.96039781209892456861945649901, 5.23744176279373040365720101496, 6.32474718318988183144196075391, 8.133622213881373387016241433677, 8.616944620973555812483758227212, 9.267063838157835864596982899938, 10.59142193972083044078893748685

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