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Properties

Label 2-352-88.51-c1-0-1
Degree $2$
Conductor $352$
Sign $-0.732 - 0.680i$
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  + (−0.852 + 2.62i)3-s + (1.35 + 1.86i)5-s + (1.08 + 3.35i)7-s + (−3.73 − 2.71i)9-s + (1.06 − 3.14i)11-s + (0.874 + 0.635i)13-s + (−6.04 + 1.96i)15-s + (0.128 + 0.176i)17-s + (−2.01 − 0.653i)19-s − 9.73·21-s − 5.68i·23-s + (−0.0920 + 0.283i)25-s + (3.60 − 2.61i)27-s + (2.14 + 6.60i)29-s + (−2.33 + 3.21i)31-s + ⋯
L(s)  = 1  + (−0.492 + 1.51i)3-s + (0.605 + 0.832i)5-s + (0.411 + 1.26i)7-s + (−1.24 − 0.903i)9-s + (0.321 − 0.946i)11-s + (0.242 + 0.176i)13-s + (−1.55 + 0.506i)15-s + (0.0311 + 0.0428i)17-s + (−0.461 − 0.149i)19-s − 2.12·21-s − 1.18i·23-s + (−0.0184 + 0.0566i)25-s + (0.693 − 0.503i)27-s + (0.398 + 1.22i)29-s + (−0.419 + 0.577i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $-0.732 - 0.680i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ -0.732 - 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.458699 + 1.16690i\)
\(L(\frac12)\) \(\approx\) \(0.458699 + 1.16690i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.06 + 3.14i)T \)
good3 \( 1 + (0.852 - 2.62i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.35 - 1.86i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.08 - 3.35i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.874 - 0.635i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.128 - 0.176i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.01 + 0.653i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.68iT - 23T^{2} \)
29 \( 1 + (-2.14 - 6.60i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.33 - 3.21i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.44 - 0.794i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.41 + 1.10i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.45iT - 43T^{2} \)
47 \( 1 + (-12.1 - 3.93i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.28 + 7.27i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.96 - 6.03i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.23 - 1.62i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 6.19T + 67T^{2} \)
71 \( 1 + (-1.31 - 1.81i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.71 - 0.557i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.386 + 0.280i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.12 - 2.92i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.99T + 89T^{2} \)
97 \( 1 + (-5.99 - 4.35i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.55059464142450271476511758855, −10.66473806484909880050839292907, −10.32263169266891216871878419890, −8.928703660020019011375248979653, −8.717062123342835198203789044036, −6.68714279065770331131764282792, −5.79934567126743077430563950636, −5.06105809115120818490714871526, −3.72486666570824065450792219133, −2.51773465552381306163268310117, 0.998440747121527647781988813123, 1.93393396180261147973891559175, 4.17399131242095572365961141552, 5.38364266023255405072822230281, 6.40599626292678851261569558554, 7.36130183497546521746571504807, 7.941329183843956647287546810882, 9.246232487014499894759913458091, 10.28265903745197167082482914465, 11.36639351398924816760187698343

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