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Properties

Label 2-294-7.2-c3-0-15
Degree $2$
Conductor $294$
Sign $-0.900 + 0.435i$
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 + (−7.94 + 13.7i)5-s + 6·6-s − 7.99·8-s + (−4.5 + 7.79i)9-s + (15.8 + 27.5i)10-s + (−28.6 − 49.7i)11-s + (6.00 − 10.3i)12-s − 5.69·13-s − 47.6·15-s + (−8 + 13.8i)16-s + (−25.9 − 44.9i)17-s + (9 + 15.5i)18-s + (8.10 − 14.0i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.711 + 1.23i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.502 + 0.870i)10-s + (−0.786 − 1.36i)11-s + (0.144 − 0.249i)12-s − 0.121·13-s − 0.821·15-s + (−0.125 + 0.216i)16-s + (−0.370 − 0.641i)17-s + (0.117 + 0.204i)18-s + (0.0978 − 0.169i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.900 + 0.435i$
Analytic conductor: \(17.3465\)
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: \((2,\ 294,\ (\ :3/2),\ -0.900 + 0.435i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5334053074\)
\(L(\frac12)\) \(\approx\) \(0.5334053074\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.94 - 13.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (28.6 + 49.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 5.69T + 2.19e3T^{2} \)
17 \( 1 + (25.9 + 44.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-8.10 + 14.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-106. + 184. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 218.T + 2.43e4T^{2} \)
31 \( 1 + (125. + 217. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (193. - 334. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 328.T + 6.89e4T^{2} \)
43 \( 1 + 37.5T + 7.95e4T^{2} \)
47 \( 1 + (127. - 220. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (105. + 183. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (206. + 356. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (418. - 724. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-82.7 - 143. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 465.T + 3.57e5T^{2} \)
73 \( 1 + (-224. - 389. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-171. + 297. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.50e3T + 5.71e5T^{2} \)
89 \( 1 + (170. - 295. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 865.T + 9.12e5T^{2} \)
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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

−11.09908840895940271767612162234, −10.34673052036421497252625922346, −9.174152020536542222692595039237, −8.125442378427555823913787385517, −7.02608986549800624459402142989, −5.77383121493717657293982455716, −4.47965397909225176127969127038, −3.27042330189544858251896286696, −2.66139985896971616853909364561, −0.16420879536044292084214607426, 1.72792034029123680582538612145, 3.61289545260754700982011518033, 4.75961861815232816516979655597, 5.58928467410931036836272388365, 7.26670133732136683151881204844, 7.61271488303290067608858973158, 8.749600082096752956540648129238, 9.450819524329009524377831763088, 10.98704800523478560119019272216, 12.22429499895160674456148382128

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