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Properties

Label 2-294-7.4-c3-0-4
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 + (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

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

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: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ -0.947 + 0.318i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.624336267\)
\(L(\frac12)\) \(\approx\) \(1.624336267\)
\(L(\frac{5}{2})\) 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

−11.76721193350968099856983026398, −10.84431217944702942816938185789, −9.925903048227719684057917470398, −9.289697357710719485078010779271, −7.55629919872492206082065598365, −7.02838830810619289338205604145, −5.81013832896537945010110333419, −5.03189498226791526889295299664, −3.59756280006545441991978242061, −2.33210916568803820367421664408, 0.55146671939718772336182736978, 1.69500779291150542998085994232, 3.20609873938642370110815888245, 4.88541248442207819400320769957, 5.54116649013508786475500050175, 6.55744131285655955994300231919, 8.344735018007855799835423166741, 8.741276838313665127515406397143, 10.12946746072709179437255402662, 10.85084304630679833494606068825

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