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Properties

Label 2-294-147.110-c1-0-11
Degree $2$
Conductor $294$
Sign $0.450 - 0.892i$
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  + (0.563 + 0.826i)2-s + (1.68 + 0.392i)3-s + (−0.365 + 0.930i)4-s + (1.89 + 0.585i)5-s + (0.626 + 1.61i)6-s + (−2.33 − 1.24i)7-s + (−0.974 + 0.222i)8-s + (2.69 + 1.32i)9-s + (0.585 + 1.89i)10-s + (0.512 − 0.0384i)11-s + (−0.981 + 1.42i)12-s + (−0.441 + 0.916i)13-s + (−0.291 − 2.62i)14-s + (2.97 + 1.73i)15-s + (−0.733 − 0.680i)16-s + (2.74 + 0.413i)17-s + ⋯
L(s)  = 1  + (0.398 + 0.584i)2-s + (0.974 + 0.226i)3-s + (−0.182 + 0.465i)4-s + (0.848 + 0.261i)5-s + (0.255 + 0.659i)6-s + (−0.883 − 0.468i)7-s + (−0.344 + 0.0786i)8-s + (0.897 + 0.440i)9-s + (0.185 + 0.600i)10-s + (0.154 − 0.0115i)11-s + (−0.283 + 0.412i)12-s + (−0.122 + 0.254i)13-s + (−0.0778 − 0.702i)14-s + (0.767 + 0.447i)15-s + (−0.183 − 0.170i)16-s + (0.666 + 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82628 + 1.12413i\)
\(L(\frac12)\) \(\approx\) \(1.82628 + 1.12413i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.563 - 0.826i)T \)
3 \( 1 + (-1.68 - 0.392i)T \)
7 \( 1 + (2.33 + 1.24i)T \)
good5 \( 1 + (-1.89 - 0.585i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (-0.512 + 0.0384i)T + (10.8 - 1.63i)T^{2} \)
13 \( 1 + (0.441 - 0.916i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (-2.74 - 0.413i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (6.52 + 3.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.417 + 2.77i)T + (-21.9 + 6.77i)T^{2} \)
29 \( 1 + (-0.493 - 0.393i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 + (0.0488 - 0.0281i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.66 - 6.77i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (1.50 + 6.61i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.323 + 1.41i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (0.902 - 0.615i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (10.9 + 4.27i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (-11.4 + 3.53i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (4.24 - 1.66i)T + (44.7 - 41.4i)T^{2} \)
67 \( 1 + (3.27 + 5.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.42 - 5.12i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (-8.41 + 12.3i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (4.00 - 6.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.9 + 6.73i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.954 - 12.7i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−12.35989513731820722584826280018, −10.69346853147547537966550802221, −9.894130612679194972280248096816, −9.105411514394823500456597241269, −8.099453974688231972466574583890, −6.89803600086454989040671721777, −6.24817117712258484950882068960, −4.70983620699232829951602282895, −3.58083197773896344629344120168, −2.36264212782492146189125207279, 1.76425727582161180477193846203, 2.90620540366522539932344361075, 4.04745044730610018919440080273, 5.62692216037737468481474399426, 6.49282981491174363421529660154, 7.935027118679488572964906588049, 9.062926861819744239086026881038, 9.693985319318040958230669916364, 10.41352463441931896622604886017, 11.91991578760297429152089814140

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