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Properties

Label 2-294-7.4-c5-0-26
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 + (2.25 + 3.89i)5-s + 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (−9.00 + 15.5i)10-s + (58.0 − 100. i)11-s + (72 + 124. i)12-s + 85.4·13-s + 40.5·15-s + (−128 − 221. i)16-s + (16.6 − 28.8i)17-s + (162 − 280. i)18-s + (317. + 550. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.0402 + 0.0697i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0284 + 0.0493i)10-s + (0.144 − 0.250i)11-s + (0.144 + 0.249i)12-s + 0.140·13-s + 0.0465·15-s + (−0.125 − 0.216i)16-s + (0.0139 − 0.0241i)17-s + (0.117 − 0.204i)18-s + (0.201 + 0.349i)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

\(L(3)\) \(\approx\) \(2.484165626\)
\(L(\frac12)\) \(\approx\) \(2.484165626\)
\(L(\frac{7}{2})\) 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 \( 1 + (-2.25 - 3.89i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-58.0 + 100. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 85.4T + 3.71e5T^{2} \)
17 \( 1 + (-16.6 + 28.8i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-317. - 550. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.36e3 + 2.36e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 5.86e3T + 2.05e7T^{2} \)
31 \( 1 + (-139. + 241. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (1.51e3 + 2.63e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 819.T + 1.15e8T^{2} \)
43 \( 1 - 1.11e4T + 1.47e8T^{2} \)
47 \( 1 + (3.70e3 + 6.41e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-6.84e3 + 1.18e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.11e4 + 1.93e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (6.34e3 + 1.09e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.61e4 + 4.52e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.02e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.84e4 + 6.66e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.67e4 - 2.90e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.05e4T + 3.93e9T^{2} \)
89 \( 1 + (-4.60e4 - 7.98e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.52e5T + 8.58e9T^{2} \)
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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

−10.92829727864364096354099214595, −9.798378957420809606178297380592, −8.610730360258950093824357658209, −7.998391760376409433992042955477, −6.80950400419099369812627654849, −6.13166889015983905626727900124, −4.84891800199306761638731825166, −3.61278462043272959497986302497, −2.34006760858001028025383182342, −0.64467589223355718018740164248, 1.16551238210768898715563918903, 2.58413574235771668159232023638, 3.68956975617950421620194180398, 4.72409667515195608625694652029, 5.74366767397527999003523320786, 7.08979245573586115360182728680, 8.373228169046046437022322483191, 9.328099407550582134273686203077, 10.08371368363493660137410922651, 11.02229594255246187111543619416

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