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Properties

Label 2-352-11.3-c1-0-7
Degree $2$
Conductor $352$
Sign $0.212 + 0.977i$
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  + (−0.780 − 2.40i)3-s + (2.30 + 1.67i)5-s + (0.780 − 2.40i)7-s + (−2.73 + 1.98i)9-s + (2.52 + 2.14i)11-s + (3.92 − 2.85i)13-s + (2.22 − 6.85i)15-s + (−4.54 − 3.30i)17-s + (0.780 + 2.40i)19-s − 6.38·21-s − 8.17·23-s + (0.972 + 2.99i)25-s + (0.780 + 0.567i)27-s + (0.427 − 1.31i)29-s + (5.35 − 3.88i)31-s + ⋯
L(s)  = 1  + (−0.450 − 1.38i)3-s + (1.03 + 0.750i)5-s + (0.295 − 0.908i)7-s + (−0.912 + 0.662i)9-s + (0.761 + 0.647i)11-s + (1.08 − 0.791i)13-s + (0.575 − 1.77i)15-s + (−1.10 − 0.800i)17-s + (0.179 + 0.551i)19-s − 1.39·21-s − 1.70·23-s + (0.194 + 0.598i)25-s + (0.150 + 0.109i)27-s + (0.0793 − 0.244i)29-s + (0.961 − 0.698i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.212 + 0.977i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.212 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11090 - 0.895564i\)
\(L(\frac12)\) \(\approx\) \(1.11090 - 0.895564i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.52 - 2.14i)T \)
good3 \( 1 + (0.780 + 2.40i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.30 - 1.67i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.780 + 2.40i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-3.92 + 2.85i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.54 + 3.30i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.780 - 2.40i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.17T + 23T^{2} \)
29 \( 1 + (-0.427 + 1.31i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.35 + 3.88i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.190 - 0.587i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.66 - 8.19i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.12T + 43T^{2} \)
47 \( 1 + (0.184 + 0.567i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.309 - 0.224i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.30 - 10.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.92 - 2.12i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + (-9.43 - 6.85i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.33 + 4.11i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.82 - 2.05i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-9.43 - 6.85i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 7.23T + 89T^{2} \)
97 \( 1 + (14.3 - 10.4i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.36488693847930734768703570863, −10.45323544454541444895031903361, −9.627876680177328023973410376934, −8.142185277465513659326682211584, −7.30011166184464454181512614819, −6.39793106243460855434006565220, −5.95396009283343828859561685685, −4.19183158978723017204706494482, −2.39249358730978230710493289303, −1.21618799661730450543885987417, 1.87426610640449237088014709512, 3.82044621435096490803343376403, 4.73912851782466611394473301457, 5.79389771833007636924653205474, 6.30934604862076730794952590361, 8.616325231783190068732858451917, 8.921612263948890093283339486247, 9.734850362993715757541245998700, 10.76219978367779286457532929348, 11.46449705451867148243816212207

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