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Properties

Label 2-294-7.4-c5-0-16
Degree $2$
Conductor $294$
Sign $0.605 + 0.795i$
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 + (−38 − 65.8i)5-s − 36·6-s + 63.9·8-s + (−40.5 − 70.1i)9-s + (−151. + 263. i)10-s + (−325 + 562. i)11-s + (72 + 124. i)12-s + 762·13-s − 684·15-s + (−128 − 221. i)16-s + (278 − 481. i)17-s + (−162 + 280. i)18-s + (1.22e3 + 2.12e3i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.679 − 1.17i)5-s − 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.480 + 0.832i)10-s + (−0.809 + 1.40i)11-s + (0.144 + 0.249i)12-s + 1.25·13-s − 0.784·15-s + (−0.125 − 0.216i)16-s + (0.233 − 0.404i)17-s + (−0.117 + 0.204i)18-s + (0.779 + 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(47.1528\)
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: \((2,\ 294,\ (\ :5/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.471982660\)
\(L(\frac12)\) \(\approx\) \(1.471982660\)
\(L(\frac{7}{2})\) 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 \( 1 + (38 + 65.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (325 - 562. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 762T + 3.71e5T^{2} \)
17 \( 1 + (-278 + 481. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.22e3 - 2.12e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.47e3 - 2.55e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 674T + 2.05e7T^{2} \)
31 \( 1 + (-1.51e3 + 2.61e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (3.86e3 + 6.69e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.70e4T + 1.15e8T^{2} \)
43 \( 1 - 2.18e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.19e4 - 2.07e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (7.79e3 - 1.35e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (2.80e3 - 4.85e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (75 + 129. i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.18e4 + 3.79e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 3.91e4T + 1.80e9T^{2} \)
73 \( 1 + (-1.17e4 + 2.04e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-8.94e3 - 1.54e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 3.89e4T + 3.93e9T^{2} \)
89 \( 1 + (3.01e3 + 5.21e3i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.08e5T + 8.58e9T^{2} \)
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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

−10.86356610235593527765081360185, −9.700036287962221582896907173827, −8.926244801914020728048263627083, −7.908477839411581696622970853979, −7.43584272782060749348703284396, −5.64067457817998490388922755269, −4.46901665326415745352033967903, −3.37415455768284197435627808878, −1.79364786778590136135815946386, −0.858699485652968570136040032823, 0.62048340714420969959255871982, 2.86278878312948385456300496138, 3.64540425112249467323085409641, 5.15787725025229725166827014531, 6.30335878424565047588493109100, 7.22594239891359400418728959938, 8.316060874130952544329382131716, 8.846083452723761958531098710979, 10.33571326127293630518893565593, 10.84929240641566422427068655942

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