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Properties

Label 2-354-59.20-c1-0-6
Degree $2$
Conductor $354$
Sign $0.986 - 0.162i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
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  + (0.907 + 0.419i)2-s + (0.947 − 0.319i)3-s + (0.647 + 0.762i)4-s + (0.401 − 0.241i)5-s + (0.994 + 0.108i)6-s + (−0.0924 − 1.70i)7-s + (0.267 + 0.963i)8-s + (0.796 − 0.605i)9-s + (0.465 − 0.0506i)10-s + (−0.0193 − 0.118i)11-s + (0.856 + 0.515i)12-s + (4.72 + 3.59i)13-s + (0.631 − 1.58i)14-s + (0.303 − 0.357i)15-s + (−0.161 + 0.986i)16-s + (0.104 − 1.92i)17-s + ⋯
L(s)  = 1  + (0.641 + 0.296i)2-s + (0.547 − 0.184i)3-s + (0.323 + 0.381i)4-s + (0.179 − 0.108i)5-s + (0.405 + 0.0441i)6-s + (−0.0349 − 0.644i)7-s + (0.0945 + 0.340i)8-s + (0.265 − 0.201i)9-s + (0.147 − 0.0160i)10-s + (−0.00583 − 0.0355i)11-s + (0.247 + 0.148i)12-s + (1.31 + 0.996i)13-s + (0.168 − 0.423i)14-s + (0.0783 − 0.0922i)15-s + (−0.0404 + 0.246i)16-s + (0.0252 − 0.466i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.986 - 0.162i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.986 - 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31671 + 0.189168i\)
\(L(\frac12)\) \(\approx\) \(2.31671 + 0.189168i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.907 - 0.419i)T \)
3 \( 1 + (-0.947 + 0.319i)T \)
59 \( 1 + (7.09 + 2.95i)T \)
good5 \( 1 + (-0.401 + 0.241i)T + (2.34 - 4.41i)T^{2} \)
7 \( 1 + (0.0924 + 1.70i)T + (-6.95 + 0.756i)T^{2} \)
11 \( 1 + (0.0193 + 0.118i)T + (-10.4 + 3.51i)T^{2} \)
13 \( 1 + (-4.72 - 3.59i)T + (3.47 + 12.5i)T^{2} \)
17 \( 1 + (-0.104 + 1.92i)T + (-16.9 - 1.83i)T^{2} \)
19 \( 1 + (2.59 + 2.45i)T + (1.02 + 18.9i)T^{2} \)
23 \( 1 + (1.36 + 0.300i)T + (20.8 + 9.65i)T^{2} \)
29 \( 1 + (7.23 - 3.34i)T + (18.7 - 22.1i)T^{2} \)
31 \( 1 + (2.74 - 2.59i)T + (1.67 - 30.9i)T^{2} \)
37 \( 1 + (-2.72 + 9.81i)T + (-31.7 - 19.0i)T^{2} \)
41 \( 1 + (3.58 - 0.789i)T + (37.2 - 17.2i)T^{2} \)
43 \( 1 + (0.304 - 1.85i)T + (-40.7 - 13.7i)T^{2} \)
47 \( 1 + (0.307 + 0.185i)T + (22.0 + 41.5i)T^{2} \)
53 \( 1 + (7.66 + 0.833i)T + (51.7 + 11.3i)T^{2} \)
61 \( 1 + (-7.17 - 3.31i)T + (39.4 + 46.4i)T^{2} \)
67 \( 1 + (0.655 + 2.36i)T + (-57.4 + 34.5i)T^{2} \)
71 \( 1 + (1.13 + 0.681i)T + (33.2 + 62.7i)T^{2} \)
73 \( 1 + (3.67 - 9.22i)T + (-52.9 - 50.2i)T^{2} \)
79 \( 1 + (-1.42 - 0.480i)T + (62.8 + 47.8i)T^{2} \)
83 \( 1 + (5.80 + 8.56i)T + (-30.7 + 77.1i)T^{2} \)
89 \( 1 + (5.62 - 2.60i)T + (57.6 - 67.8i)T^{2} \)
97 \( 1 + (-2.38 - 5.99i)T + (-70.4 + 66.7i)T^{2} \)
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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

−11.45098042864353876461746009441, −10.83016164380943609828323040569, −9.419557252333967074546564177524, −8.700153397585628614386357488141, −7.50608440556734812163067724839, −6.76836387410715969143465725939, −5.67699251658551915032006454883, −4.29965790015311677453925036699, −3.45783041079349365077796429283, −1.80436160204553479712266359581, 1.90343297900057176318161277154, 3.20502043707321776446701062589, 4.17895518379850882914326462395, 5.64249360605143564179518318569, 6.27421704652994664560570266974, 7.85587597429261553676839123577, 8.608337170973631441057059119950, 9.765579647277181349925814799903, 10.56106202773853404572126217256, 11.46408195118560656851478716998

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