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Properties

Label 2-294-7.4-c5-0-31
Degree $2$
Conductor $294$
Sign $-0.701 + 0.712i$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

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 + (13 + 22.5i)5-s + 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (−51.9 + 90.0i)10-s + (−332 + 575. i)11-s + (72 + 124. i)12-s − 318·13-s + 234·15-s + (−128 − 221. i)16-s + (791 − 1.37e3i)17-s + (162 − 280. i)18-s + (118 + 204. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.232 + 0.402i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.164 + 0.284i)10-s + (−0.827 + 1.43i)11-s + (0.144 + 0.249i)12-s − 0.521·13-s + 0.268·15-s + (−0.125 − 0.216i)16-s + (0.663 − 1.14i)17-s + (0.117 − 0.204i)18-s + (0.0749 + 0.129i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
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: \(1\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + (-13 - 22.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (332 - 575. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 318T + 3.71e5T^{2} \)
17 \( 1 + (-791 + 1.37e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-118 - 204. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.10e3 + 1.91e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 4.95e3T + 2.05e7T^{2} \)
31 \( 1 + (3.56e3 - 6.17e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.17e3 + 3.77e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.05e4T + 1.15e8T^{2} \)
43 \( 1 + 8.45e3T + 1.47e8T^{2} \)
47 \( 1 + (-2.67e3 - 4.63e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.66e4 + 2.88e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (7.71e3 - 1.33e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.83e4 + 3.18e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.04e4 - 3.54e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 9.09e3T + 1.80e9T^{2} \)
73 \( 1 + (3.67e4 - 6.36e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.47e4 + 7.74e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 6.42e3T + 3.93e9T^{2} \)
89 \( 1 + (6.13e4 + 1.06e5i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 2.13e4T + 8.58e9T^{2} \)
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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

−10.39141152760035293305931558576, −9.633333957790276435313780215183, −8.424102401585275350381456304763, −7.31588493554219713927533012591, −6.97855858514161837554408929979, −5.56266500442447780979036743548, −4.63493313928673683933317343002, −3.07888643606651900056499376836, −2.01251152983812618917090761397, 0, 1.58307967033365036755482440288, 2.99685912860729848402954950362, 3.89282050406028101637071705149, 5.27843921549101402256145001295, 5.85050155749428893350056930861, 7.66324191946719382054960701838, 8.623177262176663772063503631736, 9.525697903333520092780379414056, 10.44304000470597809147317703080

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