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Properties

Label 2-738-123.29-c1-0-12
Degree $2$
Conductor $738$
Sign $0.136 + 0.990i$
Analytic cond. $5.89295$
Root an. cond. $2.42754$
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  + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (−1.48 − 2.90i)5-s + (2.16 − 1.32i)7-s + (0.891 + 0.453i)8-s + (−1.00 − 3.10i)10-s + (−3.34 − 0.263i)11-s + (−0.135 − 0.563i)13-s + (2.34 − 0.972i)14-s + (0.809 + 0.587i)16-s + (−1.23 − 1.44i)17-s + (1.37 − 5.72i)19-s + (−0.510 − 3.22i)20-s + (−3.26 − 0.782i)22-s + (−1.86 + 1.35i)23-s + ⋯
L(s)  = 1  + (0.698 + 0.110i)2-s + (0.475 + 0.154i)4-s + (−0.662 − 1.30i)5-s + (0.818 − 0.501i)7-s + (0.315 + 0.160i)8-s + (−0.318 − 0.981i)10-s + (−1.00 − 0.0793i)11-s + (−0.0375 − 0.156i)13-s + (0.627 − 0.259i)14-s + (0.202 + 0.146i)16-s + (−0.300 − 0.351i)17-s + (0.315 − 1.31i)19-s + (−0.114 − 0.720i)20-s + (−0.695 − 0.166i)22-s + (−0.388 + 0.281i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(738\)    =    \(2 \cdot 3^{2} \cdot 41\)
Sign: $0.136 + 0.990i$
Analytic conductor: \(5.89295\)
Root analytic conductor: \(2.42754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{738} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 738,\ (\ :1/2),\ 0.136 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48623 - 1.29599i\)
\(L(\frac12)\) \(\approx\) \(1.48623 - 1.29599i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 - 0.156i)T \)
3 \( 1 \)
41 \( 1 + (-0.432 - 6.38i)T \)
good5 \( 1 + (1.48 + 2.90i)T + (-2.93 + 4.04i)T^{2} \)
7 \( 1 + (-2.16 + 1.32i)T + (3.17 - 6.23i)T^{2} \)
11 \( 1 + (3.34 + 0.263i)T + (10.8 + 1.72i)T^{2} \)
13 \( 1 + (0.135 + 0.563i)T + (-11.5 + 5.90i)T^{2} \)
17 \( 1 + (1.23 + 1.44i)T + (-2.65 + 16.7i)T^{2} \)
19 \( 1 + (-1.37 + 5.72i)T + (-16.9 - 8.62i)T^{2} \)
23 \( 1 + (1.86 - 1.35i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-4.84 + 5.67i)T + (-4.53 - 28.6i)T^{2} \)
31 \( 1 + (-0.557 + 0.181i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.294 - 0.907i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (0.642 - 4.05i)T + (-40.8 - 13.2i)T^{2} \)
47 \( 1 + (-4.74 + 7.74i)T + (-21.3 - 41.8i)T^{2} \)
53 \( 1 + (-1.55 - 1.32i)T + (8.29 + 52.3i)T^{2} \)
59 \( 1 + (5.35 + 7.36i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-14.3 + 2.27i)T + (58.0 - 18.8i)T^{2} \)
67 \( 1 + (-4.21 + 0.331i)T + (66.1 - 10.4i)T^{2} \)
71 \( 1 + (0.665 - 8.45i)T + (-70.1 - 11.1i)T^{2} \)
73 \( 1 + (-3.72 + 3.72i)T - 73iT^{2} \)
79 \( 1 + (-2.04 - 0.847i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 - 6.19iT - 83T^{2} \)
89 \( 1 + (-4.49 - 7.33i)T + (-40.4 + 79.2i)T^{2} \)
97 \( 1 + (-0.784 - 9.97i)T + (-95.8 + 15.1i)T^{2} \)
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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

−10.33368823716000572046839771113, −9.205043549350937661509035433144, −8.097751200880510397989108984216, −7.84637402859106874857050863074, −6.65532675635875858191185271505, −5.21525853512709015403132650803, −4.84952731867560688880650026958, −3.97610379317987218294862027202, −2.51447918875923384016243411136, −0.819503139419171617795848401545, 2.04897987100817026317875389555, 3.04912576884442447762700571312, 4.04452487243293116347539091578, 5.15928206177942664981899232495, 6.06106747648268551942262473659, 7.10430861352584334348644217954, 7.80135888177967210686186297779, 8.632211483373877628668030932659, 10.22784314273390272924849706443, 10.61380531255212971909850631295

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