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Properties

Label 2-294-7.4-c5-0-19
Degree $2$
Conductor $294$
Sign $0.900 + 0.435i$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)2-s + (−4.5 + 7.79i)3-s + (−7.99 + 13.8i)4-s + (30.5 + 52.8i)5-s + 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (122. − 211. i)10-s + (18.2 − 31.6i)11-s + (−72 − 124. i)12-s + 34.5·13-s − 549.·15-s + (−128 − 221. i)16-s + (1.03e3 − 1.78e3i)17-s + (−162 + 280. i)18-s + (−226. − 391. i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.546 + 0.946i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.386 − 0.669i)10-s + (0.0455 − 0.0788i)11-s + (−0.144 − 0.249i)12-s + 0.0567·13-s − 0.630·15-s + (−0.125 − 0.216i)16-s + (0.864 − 1.49i)17-s + (−0.117 + 0.204i)18-s + (−0.143 − 0.248i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ 0.900 + 0.435i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.549233644\)
\(L(\frac12)\) \(\approx\) \(1.549233644\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 3.46i)T \)
3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (-30.5 - 52.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-18.2 + 31.6i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 34.5T + 3.71e5T^{2} \)
17 \( 1 + (-1.03e3 + 1.78e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (226. + 391. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (842. + 1.45e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 4.76e3T + 2.05e7T^{2} \)
31 \( 1 + (-2.63e3 + 4.55e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-6.41e3 - 1.11e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 7.12e3T + 1.15e8T^{2} \)
43 \( 1 - 1.11e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.17e4 - 2.03e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-3.51e3 + 6.08e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.21e4 + 3.82e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (9.69e3 + 1.67e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.04e4 - 1.81e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 7.98e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.85e4 + 3.20e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.10e4 + 3.64e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.31e3T + 3.93e9T^{2} \)
89 \( 1 + (-2.57e4 - 4.45e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.27e5T + 8.58e9T^{2} \)
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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

−10.87923620233826606212448345331, −9.873576427387013293876501970022, −9.469139069364412359603211788551, −8.088016442433662616360030767258, −6.95685814746379592541523498874, −5.88214388309627707603379116910, −4.63544094189059521318662516656, −3.28212807909080380237813309844, −2.36027543222915141452321961214, −0.64728673517564259845963850735, 0.920144686825478246991350279564, 1.87309944060745736984388791615, 3.97246458168992858530276665448, 5.42929931438859363164514173476, 5.89305400142353704721752748212, 7.16303003550840066179163364602, 8.137161832933176238838386454617, 8.935348021224542482968687719946, 9.911902960021351977576981427562, 10.84510738049131759334532256222

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