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Properties

Label 2-352-11.5-c1-0-5
Degree $2$
Conductor $352$
Sign $0.998 - 0.0475i$
Analytic cond. $2.81073$
Root an. cond. $1.67652$
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 + 0.363i)3-s + (−0.381 + 1.17i)5-s + (3.61 − 2.62i)7-s + (−0.809 − 2.48i)9-s + (2.19 + 2.48i)11-s + (−1 − 3.07i)13-s + (−0.618 + 0.449i)15-s + (−1.5 + 4.61i)17-s + (6.16 + 4.47i)19-s + 2.76·21-s + 0.763·23-s + (2.80 + 2.04i)25-s + (1.07 − 3.30i)27-s + (−0.381 + 0.277i)29-s + (−2.61 − 8.05i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.209i)3-s + (−0.170 + 0.525i)5-s + (1.36 − 0.993i)7-s + (−0.269 − 0.829i)9-s + (0.660 + 0.750i)11-s + (−0.277 − 0.853i)13-s + (−0.159 + 0.115i)15-s + (−0.363 + 1.11i)17-s + (1.41 + 1.02i)19-s + 0.603·21-s + 0.159·23-s + (0.561 + 0.408i)25-s + (0.206 − 0.635i)27-s + (−0.0709 + 0.0515i)29-s + (−0.470 − 1.44i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.998 - 0.0475i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.998 - 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62402 + 0.0386167i\)
\(L(\frac12)\) \(\approx\) \(1.62402 + 0.0386167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-2.19 - 2.48i)T \)
good3 \( 1 + (-0.5 - 0.363i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.381 - 1.17i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.61 + 2.62i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1 + 3.07i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.5 - 4.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.16 - 4.47i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.763T + 23T^{2} \)
29 \( 1 + (0.381 - 0.277i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.61 + 8.05i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.85 - 2.80i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.73 + 4.89i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.09T + 43T^{2} \)
47 \( 1 + (-2 - 1.45i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.763 + 2.35i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.69 - 1.22i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2 - 6.15i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 3.61T + 67T^{2} \)
71 \( 1 + (3 - 9.23i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.73 - 4.16i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.85 + 5.70i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.42 - 4.39i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 8.61T + 89T^{2} \)
97 \( 1 + (-1.04 - 3.21i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.47359225357201708189031753941, −10.51021775585035140639259775096, −9.818492292597473859756876599592, −8.581105219113441265979541072088, −7.69152596309609701964949988696, −6.92647126708078113529242683058, −5.54532072691975038745357386987, −4.25270029778607220720638691903, −3.39196203021968693489357201525, −1.49725199424776432196193796858, 1.60178399978884122908616973977, 2.93155851685162470575868904004, 4.88162484982752203680961607332, 5.15774658747839462662738935117, 6.82781049533823022852340650518, 7.86334053196373351174900518745, 8.837347913258328004729102673448, 9.113699778212764753411329746428, 10.83895021703629983305774333074, 11.67621072211849143660359371315

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