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Properties

Label 2-294-147.101-c1-0-9
Degree $2$
Conductor $294$
Sign $0.521 - 0.853i$
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  + (0.930 + 0.365i)2-s + (1.14 + 1.30i)3-s + (0.733 + 0.680i)4-s + (−1.06 − 0.728i)5-s + (0.586 + 1.62i)6-s + (2.60 + 0.459i)7-s + (0.433 + 0.900i)8-s + (−0.395 + 2.97i)9-s + (−0.728 − 1.06i)10-s + (−0.431 − 2.86i)11-s + (−0.0495 + 1.73i)12-s + (−2.21 + 1.76i)13-s + (2.25 + 1.37i)14-s + (−0.270 − 2.22i)15-s + (0.0747 + 0.997i)16-s + (−1.26 − 0.389i)17-s + ⋯
L(s)  = 1  + (0.658 + 0.258i)2-s + (0.658 + 0.752i)3-s + (0.366 + 0.340i)4-s + (−0.478 − 0.325i)5-s + (0.239 + 0.665i)6-s + (0.984 + 0.173i)7-s + (0.153 + 0.318i)8-s + (−0.131 + 0.991i)9-s + (−0.230 − 0.338i)10-s + (−0.130 − 0.863i)11-s + (−0.0143 + 0.499i)12-s + (−0.613 + 0.489i)13-s + (0.603 + 0.368i)14-s + (−0.0698 − 0.574i)15-s + (0.0186 + 0.249i)16-s + (−0.305 − 0.0943i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.521 - 0.853i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91081 + 1.07188i\)
\(L(\frac12)\) \(\approx\) \(1.91081 + 1.07188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.930 - 0.365i)T \)
3 \( 1 + (-1.14 - 1.30i)T \)
7 \( 1 + (-2.60 - 0.459i)T \)
good5 \( 1 + (1.06 + 0.728i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (0.431 + 2.86i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (2.21 - 1.76i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (1.26 + 0.389i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-2.16 + 1.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.228 - 0.739i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-3.46 + 0.789i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (6.98 + 4.03i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.83 + 3.56i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (7.24 - 3.48i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (1.28 + 0.620i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.02 + 2.62i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-7.58 + 8.17i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-8.45 + 5.76i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-0.237 - 0.255i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (7.17 - 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.66 - 1.29i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (6.12 - 2.40i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (4.93 + 8.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.79 - 4.75i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (14.4 + 2.18i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 7.81iT - 97T^{2} \)
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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

−11.70325346750031997990359245653, −11.29775295191655253058844901196, −10.06589950144479745442065676730, −8.828060748925264699011510070108, −8.188530077530774396472670914979, −7.23250754486357220401389778358, −5.56580698911612709425579670795, −4.70585810671350803228497246208, −3.78964084184292189779458221205, −2.37336801136592222960142076962, 1.68798406366650974041615185192, 2.98975532996915063132024348180, 4.26766082880727523069579094975, 5.46519380439375208855425631515, 7.02721489465075174806823774261, 7.51652959179618346798182782740, 8.559625862328503928280184309515, 9.859338722478449130971504631620, 10.91619932913927385218675411154, 11.90049493981319078579761392325

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