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Properties

Label 2-294-7.2-c3-0-8
Degree $2$
Conductor $294$
Sign $0.947 + 0.318i$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (−3.70 + 6.42i)5-s + 6·6-s + 7.99·8-s + (−4.5 + 7.79i)9-s + (−7.41 − 12.8i)10-s + (−5.24 − 9.08i)11-s + (−6.00 + 10.3i)12-s + 2.78·13-s + 22.2·15-s + (−8 + 13.8i)16-s + (−25.2 − 43.6i)17-s + (−9 − 15.5i)18-s + (−62.5 + 108. i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.331 + 0.574i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.234 − 0.406i)10-s + (−0.143 − 0.248i)11-s + (−0.144 + 0.249i)12-s + 0.0594·13-s + 0.382·15-s + (−0.125 + 0.216i)16-s + (−0.359 − 0.623i)17-s + (−0.117 − 0.204i)18-s + (−0.754 + 1.30i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 0.947 + 0.318i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.029471977\)
\(L(\frac12)\) \(\approx\) \(1.029471977\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 + (1.5 + 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (3.70 - 6.42i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (5.24 + 9.08i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 2.78T + 2.19e3T^{2} \)
17 \( 1 + (25.2 + 43.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (62.5 - 108. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-91.1 + 157. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 156.T + 2.43e4T^{2} \)
31 \( 1 + (69.8 + 120. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-197. + 341. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 197.T + 6.89e4T^{2} \)
43 \( 1 - 343.T + 7.95e4T^{2} \)
47 \( 1 + (-305. + 528. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-68.7 - 119. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (294. + 510. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (123. - 214. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-197. - 342. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 285.T + 3.57e5T^{2} \)
73 \( 1 + (-498. - 863. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-424. + 734. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 210.T + 5.71e5T^{2} \)
89 \( 1 + (-276. + 479. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 903.T + 9.12e5T^{2} \)
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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

−11.05202399713022716756008626010, −10.54276319241293598506965001637, −9.214630248428492358742622103968, −8.238633260271698171702504594253, −7.33450809360205102366023042612, −6.52433066567771156644882290009, −5.56434141907230460663369170930, −4.15919475010250069959434393158, −2.46212399574432505752428575195, −0.59439002931141291112603504723, 0.988309156185134890350063563244, 2.76871098709291140076328244499, 4.20237759534950672199099351072, 4.97365926057605728641642837716, 6.46034210490824823409009800311, 7.77106801055823858706039026568, 8.824927097853212319369057582837, 9.446519981137729457027028248338, 10.65078792304278834503434568515, 11.17419476626184135281910202643

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