Properties

Label 2-294-7.4-c3-0-11
Degree $2$
Conductor $294$
Sign $0.605 + 0.795i$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
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 − 1.73i)2-s + (−1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (−1 − 1.73i)5-s + 6·6-s + 7.99·8-s + (−4.5 − 7.79i)9-s + (−1.99 + 3.46i)10-s + (4 − 6.92i)11-s + (−6.00 − 10.3i)12-s − 42·13-s + 6·15-s + (−8 − 13.8i)16-s + (1 − 1.73i)17-s + (−9 + 15.5i)18-s + (62 + 107. i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.0894 − 0.154i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0632 + 0.109i)10-s + (0.109 − 0.189i)11-s + (−0.144 − 0.249i)12-s − 0.896·13-s + 0.103·15-s + (−0.125 − 0.216i)16-s + (0.0142 − 0.0247i)17-s + (−0.117 + 0.204i)18-s + (0.748 + 1.29i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.145887577\)
\(L(\frac12)\) \(\approx\) \(1.145887577\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (1 + 1.73i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-4 + 6.92i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 42T + 2.19e3T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-62 - 107. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (38 + 65.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 254T + 2.43e4T^{2} \)
31 \( 1 + (-36 + 62.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (199 + 344. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 462T + 6.89e4T^{2} \)
43 \( 1 - 212T + 7.95e4T^{2} \)
47 \( 1 + (-132 - 228. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-81 + 140. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-386 + 668. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (15 + 25.9i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-382 + 661. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 236T + 3.57e5T^{2} \)
73 \( 1 + (209 - 361. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (276 + 478. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.03e3T + 5.71e5T^{2} \)
89 \( 1 + (15 + 25.9i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.19e3T + 9.12e5T^{2} \)
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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

−11.07443766045306164027814323012, −10.25539281831255275006967577502, −9.551528381142923962546914405606, −8.511891307323461194799340052919, −7.53714887398862525827833954880, −6.14297698309777683964648137273, −4.89388587069008611216404283598, −3.85206920583382157080152735104, −2.47697824517564855148506211167, −0.66895836114772020108419299615, 0.986466579720215026231979625906, 2.74173493525254529146793396765, 4.61955876626403286174948777698, 5.59358524038180540168885129591, 6.87328990228092010242518390699, 7.34712297921360505520584870228, 8.509382046497376126155616272364, 9.500653316873998047545419363587, 10.44686246803069579045987689704, 11.52727463213481976172045704171

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