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Properties

Label 2-294-21.17-c1-0-10
Degree $2$
Conductor $294$
Sign $-0.607 + 0.794i$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−1.73 + 0.0675i)3-s + (0.499 − 0.866i)4-s + (−0.765 − 1.32i)5-s + (−1.46 + 0.923i)6-s − 0.999i·8-s + (2.99 − 0.233i)9-s + (−1.32 − 0.765i)10-s + (−5.40 − 3.12i)11-s + (−0.806 + 1.53i)12-s − 5.22i·13-s + (1.41 + 2.24i)15-s + (−0.5 − 0.866i)16-s + (−1.14 + 1.98i)17-s + (2.47 − 1.69i)18-s + (1.60 − 0.923i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.999 + 0.0390i)3-s + (0.249 − 0.433i)4-s + (−0.342 − 0.592i)5-s + (−0.598 + 0.377i)6-s − 0.353i·8-s + (0.996 − 0.0779i)9-s + (−0.419 − 0.242i)10-s + (−1.63 − 0.941i)11-s + (−0.232 + 0.442i)12-s − 1.44i·13-s + (0.365 + 0.579i)15-s + (−0.125 − 0.216i)16-s + (−0.278 + 0.482i)17-s + (0.582 − 0.400i)18-s + (0.367 − 0.211i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.607 + 0.794i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ -0.607 + 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431150 - 0.872668i\)
\(L(\frac12)\) \(\approx\) \(0.431150 - 0.872668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (1.73 - 0.0675i)T \)
7 \( 1 \)
good5 \( 1 + (0.765 + 1.32i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.40 + 3.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.22iT - 13T^{2} \)
17 \( 1 + (1.14 - 1.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.60 + 0.923i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.44 - 1.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.828iT - 29T^{2} \)
31 \( 1 + (-5.85 - 3.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.23T + 41T^{2} \)
43 \( 1 - 5.07T + 43T^{2} \)
47 \( 1 + (1.84 + 3.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.61 - 3.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.23 + 3.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.07 - 2.93i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.24 - 5.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.17iT - 71T^{2} \)
73 \( 1 + (-0.662 - 0.382i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.58 + 7.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.131T + 83T^{2} \)
89 \( 1 + (3.08 + 5.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.48iT - 97T^{2} \)
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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

−11.47479879442928887605168024633, −10.63482591094044951365335472271, −10.09873615317837801470643396881, −8.440376153509638282689749362404, −7.54461738939322655410301824494, −5.99227494430771273348900835620, −5.39430397447846717157746247544, −4.40544683691320161036530539928, −2.90758371008446805107715495234, −0.66291660493181740856005051488, 2.41262570323153091161121820616, 4.20000823871167028796653801911, 5.01849821484223882589364527818, 6.17718660414199758501429932948, 7.14437476866676106715011694415, 7.74209192852743763592388678478, 9.474878691319813924678072244100, 10.51825979256774128880774019359, 11.31078211973144876585083145879, 12.09578299689025431339113108771

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