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Properties

Label 2-294-49.16-c1-0-3
Degree $2$
Conductor $294$
Sign $-0.815 - 0.578i$
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  + (0.365 + 0.930i)2-s + (0.0747 + 0.997i)3-s + (−0.733 + 0.680i)4-s + (−0.549 + 0.374i)5-s + (−0.900 + 0.433i)6-s + (−0.0527 + 2.64i)7-s + (−0.900 − 0.433i)8-s + (−0.988 + 0.149i)9-s + (−0.549 − 0.374i)10-s + (−0.462 − 0.0697i)11-s + (−0.733 − 0.680i)12-s + (−1.08 + 1.35i)13-s + (−2.48 + 0.917i)14-s + (−0.415 − 0.520i)15-s + (0.0747 − 0.997i)16-s + (0.323 − 0.0999i)17-s + ⋯
L(s)  = 1  + (0.258 + 0.658i)2-s + (0.0431 + 0.575i)3-s + (−0.366 + 0.340i)4-s + (−0.245 + 0.167i)5-s + (−0.367 + 0.177i)6-s + (−0.0199 + 0.999i)7-s + (−0.318 − 0.153i)8-s + (−0.329 + 0.0496i)9-s + (−0.173 − 0.118i)10-s + (−0.139 − 0.0210i)11-s + (−0.211 − 0.196i)12-s + (−0.300 + 0.376i)13-s + (−0.663 + 0.245i)14-s + (−0.107 − 0.134i)15-s + (0.0186 − 0.249i)16-s + (0.0785 − 0.0242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.815 - 0.578i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ -0.815 - 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371263 + 1.16522i\)
\(L(\frac12)\) \(\approx\) \(0.371263 + 1.16522i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.365 - 0.930i)T \)
3 \( 1 + (-0.0747 - 0.997i)T \)
7 \( 1 + (0.0527 - 2.64i)T \)
good5 \( 1 + (0.549 - 0.374i)T + (1.82 - 4.65i)T^{2} \)
11 \( 1 + (0.462 + 0.0697i)T + (10.5 + 3.24i)T^{2} \)
13 \( 1 + (1.08 - 1.35i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.323 + 0.0999i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (0.817 - 1.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.81 - 1.17i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (-2.18 + 9.59i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-2.92 - 5.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.99 - 1.84i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (0.571 + 0.275i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (-10.8 + 5.22i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-0.598 - 1.52i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (-2.76 + 2.56i)T + (3.96 - 52.8i)T^{2} \)
59 \( 1 + (-7.33 - 5.00i)T + (21.5 + 54.9i)T^{2} \)
61 \( 1 + (-6.15 - 5.71i)T + (4.55 + 60.8i)T^{2} \)
67 \( 1 + (-4.09 - 7.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.94 + 8.51i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (0.876 - 2.23i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (-5.65 + 9.78i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.87 + 4.85i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (13.9 - 2.09i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + 0.540T + 97T^{2} \)
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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

−12.08612559871856576638796452499, −11.39206870281967487288896102999, −10.11094887088120005583253585967, −9.185250762856421370218045670402, −8.377356097053543857401472942907, −7.29527130404580101089330518893, −6.07879791928222934683362550432, −5.19338542324672185632384705998, −4.04521409180437704222479295043, −2.68229587221740320667942428720, 0.874300838984387072748639120791, 2.64889145592788584486734730348, 3.99685042391175331517372449190, 5.11439510728937624163096016185, 6.52716543146777534818444208056, 7.53039176435934340487199433469, 8.507651612369510867678074020420, 9.702360077355089418115055473269, 10.66830084423195702211136458306, 11.37029582568237923081639848368

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