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Properties

Label 2-294-7.4-c5-0-12
Degree $2$
Conductor $294$
Sign $0.386 - 0.922i$
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 + (11.2 + 19.4i)5-s + 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (44.9 − 77.9i)10-s + (−170. + 294. i)11-s + (−72 − 124. i)12-s + 728.·13-s − 202.·15-s + (−128 − 221. i)16-s + (404. − 701. i)17-s + (−162 + 280. i)18-s + (513. + 888. i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.201 + 0.348i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.142 − 0.246i)10-s + (−0.424 + 0.734i)11-s + (−0.144 − 0.249i)12-s + 1.19·13-s − 0.232·15-s + (−0.125 − 0.216i)16-s + (0.339 − 0.588i)17-s + (−0.117 + 0.204i)18-s + (0.326 + 0.564i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
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: \((2,\ 294,\ (\ :5/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.328905147\)
\(L(\frac12)\) \(\approx\) \(1.328905147\)
\(L(\frac{7}{2})\) 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 \( 1 + (-11.2 - 19.4i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (170. - 294. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 728.T + 3.71e5T^{2} \)
17 \( 1 + (-404. + 701. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-513. - 888. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (711. + 1.23e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 5.21e3T + 2.05e7T^{2} \)
31 \( 1 + (3.51e3 - 6.09e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (6.39e3 + 1.10e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.17e3T + 1.15e8T^{2} \)
43 \( 1 - 3.66e3T + 1.47e8T^{2} \)
47 \( 1 + (-4.65e3 - 8.06e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.78e4 - 3.08e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.51e4 - 2.63e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.60e4 - 2.78e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.06e4 - 1.84e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.11e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.06e4 + 3.57e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.75e4 - 3.03e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 8.61e4T + 3.93e9T^{2} \)
89 \( 1 + (-3.89e4 - 6.75e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.61e5T + 8.58e9T^{2} \)
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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

−10.71673538095802339354820289310, −10.46898081972947847047049845440, −9.391034478897736813604934927572, −8.492095190300712325178801568986, −7.31594869983354040218129889810, −6.11807911051165345421185958271, −4.89996542908997233300932179086, −3.73973190031628130907894059642, −2.57326640511419660239006376063, −1.08414002896829867392297372945, 0.50703506543731530270446824941, 1.60485374640931317454032050579, 3.42034549365550281877376625769, 5.04821341870029481448958778208, 5.91126277404953145765458783994, 6.75756850157364672983408263036, 7.996093207311133050971425787702, 8.579720729718251855214782708841, 9.678637711320721846694431433760, 10.80027010386463807244771222791

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