Properties

Label 2-294-147.110-c1-0-12
Degree $2$
Conductor $294$
Sign $0.881 - 0.471i$
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 + (0.634 − 1.61i)3-s + (−0.365 + 0.930i)4-s + (3.89 + 1.20i)5-s + (1.68 − 0.383i)6-s + (0.170 + 2.64i)7-s + (−0.974 + 0.222i)8-s + (−2.19 − 2.04i)9-s + (1.20 + 3.89i)10-s + (−4.55 + 0.341i)11-s + (1.26 + 1.17i)12-s + (1.44 − 2.99i)13-s + (−2.08 + 1.62i)14-s + (4.40 − 5.51i)15-s + (−0.733 − 0.680i)16-s + (−0.806 − 0.121i)17-s + ⋯
L(s)  = 1  + (0.398 + 0.584i)2-s + (0.366 − 0.930i)3-s + (−0.182 + 0.465i)4-s + (1.74 + 0.537i)5-s + (0.689 − 0.156i)6-s + (0.0645 + 0.997i)7-s + (−0.344 + 0.0786i)8-s + (−0.731 − 0.681i)9-s + (0.379 + 1.23i)10-s + (−1.37 + 0.102i)11-s + (0.366 + 0.340i)12-s + (0.400 − 0.831i)13-s + (−0.557 + 0.435i)14-s + (1.13 − 1.42i)15-s + (−0.183 − 0.170i)16-s + (−0.195 − 0.0294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.881 - 0.471i$
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: \((2,\ 294,\ (\ :1/2),\ 0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97998 + 0.496619i\)
\(L(\frac12)\) \(\approx\) \(1.97998 + 0.496619i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.634 + 1.61i)T \)
7 \( 1 + (-0.170 - 2.64i)T \)
good5 \( 1 + (-3.89 - 1.20i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (4.55 - 0.341i)T + (10.8 - 1.63i)T^{2} \)
13 \( 1 + (-1.44 + 2.99i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (0.806 + 0.121i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-0.568 - 0.327i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.306 + 2.03i)T + (-21.9 + 6.77i)T^{2} \)
29 \( 1 + (2.33 + 1.85i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 + (-3.95 + 2.28i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.44 + 3.69i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (2.36 + 10.3i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.407 + 1.78i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (10.8 - 7.37i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-6.90 - 2.71i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (9.88 - 3.04i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (-9.28 + 3.64i)T + (44.7 - 41.4i)T^{2} \)
67 \( 1 + (-4.09 - 7.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.00 - 3.19i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (6.33 - 9.29i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (0.703 - 1.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.4 + 5.97i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.501 - 6.69i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−12.28455974369916511990396460921, −10.89179528869541753306400679967, −9.830513525875356253322136844311, −8.810759166737797577348324441993, −7.925450974814859435809372991640, −6.77621535594776157596211552072, −5.81182081024182197092081467537, −5.41183527428152341847979386394, −2.91979536256857586894514931104, −2.18705669287700144544896411990, 1.80154032172172566931877281252, 3.14514759521738886607365747364, 4.63969207334389447320959329533, 5.26713142428880263636110754290, 6.45788513154491854046017809082, 8.197978845030156868518054664604, 9.247491701460976591773295523881, 10.02670377058878849783165358642, 10.47530574492700085330510498108, 11.45863306031003890415956059239

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