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Properties

Label 2-396-44.43-c1-0-11
Degree $2$
Conductor $396$
Sign $0.976 + 0.216i$
Analytic cond. $3.16207$
Root an. cond. $1.77822$
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  + (−1.19 − 0.750i)2-s + (0.874 + 1.79i)4-s + 3.34·5-s + 3.56·7-s + (0.301 − 2.81i)8-s + (−4.01 − 2.51i)10-s + (−2.06 + 2.59i)11-s − 0.896i·13-s + (−4.27 − 2.67i)14-s + (−2.47 + 3.14i)16-s + 7.46i·17-s − 1.45·19-s + (2.92 + 6.01i)20-s + (4.42 − 1.56i)22-s − 3.34i·23-s + ⋯
L(s)  = 1  + (−0.847 − 0.530i)2-s + (0.437 + 0.899i)4-s + 1.49·5-s + 1.34·7-s + (0.106 − 0.994i)8-s + (−1.26 − 0.793i)10-s + (−0.621 + 0.783i)11-s − 0.248i·13-s + (−1.14 − 0.714i)14-s + (−0.617 + 0.786i)16-s + 1.80i·17-s − 0.334·19-s + (0.653 + 1.34i)20-s + (0.942 − 0.334i)22-s − 0.697i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(396\)    =    \(2^{2} \cdot 3^{2} \cdot 11\)
Sign: $0.976 + 0.216i$
Analytic conductor: \(3.16207\)
Root analytic conductor: \(1.77822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{396} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 396,\ (\ :1/2),\ 0.976 + 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29117 - 0.141690i\)
\(L(\frac12)\) \(\approx\) \(1.29117 - 0.141690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.750i)T \)
3 \( 1 \)
11 \( 1 + (2.06 - 2.59i)T \)
good5 \( 1 - 3.34T + 5T^{2} \)
7 \( 1 - 3.56T + 7T^{2} \)
13 \( 1 + 0.896iT - 13T^{2} \)
17 \( 1 - 7.46iT - 17T^{2} \)
19 \( 1 + 1.45T + 19T^{2} \)
23 \( 1 + 3.34iT - 23T^{2} \)
29 \( 1 + 6.25iT - 29T^{2} \)
31 \( 1 + 3.49iT - 31T^{2} \)
37 \( 1 + 7.19T + 37T^{2} \)
41 \( 1 - 3.78iT - 41T^{2} \)
43 \( 1 - 2.12T + 43T^{2} \)
47 \( 1 + 4.84iT - 47T^{2} \)
53 \( 1 - 7.04T + 53T^{2} \)
59 \( 1 + 1.19iT - 59T^{2} \)
61 \( 1 + 1.56iT - 61T^{2} \)
67 \( 1 + 3.19iT - 67T^{2} \)
71 \( 1 + 5.84iT - 71T^{2} \)
73 \( 1 - 8.46iT - 73T^{2} \)
79 \( 1 - 5.35T + 79T^{2} \)
83 \( 1 - 4.20T + 83T^{2} \)
89 \( 1 + 1.69T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−10.84831869113299745918353204237, −10.39108586467992778191880094309, −9.624346093915869605158162877823, −8.496602987878611723667012867353, −7.919575298676593096186692018069, −6.61202908319699811260427105760, −5.52164723366301717920227879006, −4.24761307741557911824411275793, −2.33021999964152920580161835304, −1.67405622681042615483150357348, 1.39095049556386140512686088303, 2.55695724596299326463836153577, 5.12935528177724879010098814236, 5.42547297014829347997382236569, 6.71544483215431992625943526296, 7.64394827704444499920708099864, 8.734555713216642756868441220524, 9.298350990142765958405176162129, 10.37683206845806038299800373469, 10.96903593079659646756243807330

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