Properties

Label 2-294-7.4-c1-0-1
Degree $2$
Conductor $294$
Sign $-0.605 - 0.795i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (2 + 3.46i)5-s − 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.99 + 3.46i)10-s + (2 − 3.46i)11-s + (−0.499 − 0.866i)12-s − 4·13-s − 3.99·15-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (2 + 3.46i)19-s − 3.99·20-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.894 + 1.54i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.632 + 1.09i)10-s + (0.603 − 1.04i)11-s + (−0.144 − 0.249i)12-s − 1.10·13-s − 1.03·15-s + (−0.125 − 0.216i)16-s + (0.117 − 0.204i)18-s + (0.458 + 0.794i)19-s − 0.894·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.668907 + 1.34934i\)
\(L(\frac12)\) \(\approx\) \(0.668907 + 1.34934i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (8 - 13.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−11.92600478653742455092801157842, −11.18722997284546243693424283075, −10.09261000927074401530189068795, −9.566876486194536292856387732200, −8.118127987186609001252455802406, −6.90712420659255009124626991933, −6.19755467608876817418593479644, −5.34647480363494589389057159411, −3.78289372330496352835179715741, −2.66643210375454565960710431043, 1.16128895222006788874837597927, 2.37496962062192718155922833305, 4.55267023697914273424721480030, 5.08433840649501812970277510919, 6.27130054339859123027138020953, 7.53080043398771060766144080621, 8.976149509284075134878791999192, 9.501239565118286709381648977725, 10.47266445787759645887529501826, 11.97888609746625118435216627925

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