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Properties

Label 2-352-11.4-c1-0-0
Degree $2$
Conductor $352$
Sign $-0.919 + 0.392i$
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  + (−0.945 + 2.90i)3-s + (−2.43 + 1.76i)5-s + (0.483 + 1.48i)7-s + (−5.14 − 3.73i)9-s + (3.21 + 0.815i)11-s + (−2.95 − 2.14i)13-s + (−2.84 − 8.74i)15-s + (3.62 − 2.63i)17-s + (−0.848 + 2.61i)19-s − 4.78·21-s − 4.77·23-s + (1.24 − 3.83i)25-s + (8.31 − 6.03i)27-s + (1.53 + 4.72i)29-s + (0.394 + 0.286i)31-s + ⋯
L(s)  = 1  + (−0.545 + 1.67i)3-s + (−1.08 + 0.790i)5-s + (0.182 + 0.562i)7-s + (−1.71 − 1.24i)9-s + (0.969 + 0.245i)11-s + (−0.820 − 0.596i)13-s + (−0.733 − 2.25i)15-s + (0.878 − 0.638i)17-s + (−0.194 + 0.599i)19-s − 1.04·21-s − 0.995·23-s + (0.249 − 0.767i)25-s + (1.59 − 1.16i)27-s + (0.285 + 0.877i)29-s + (0.0708 + 0.0514i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $-0.919 + 0.392i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ -0.919 + 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.124270 - 0.607467i\)
\(L(\frac12)\) \(\approx\) \(0.124270 - 0.607467i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.21 - 0.815i)T \)
good3 \( 1 + (0.945 - 2.90i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.43 - 1.76i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.483 - 1.48i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.95 + 2.14i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.62 + 2.63i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.848 - 2.61i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.77T + 23T^{2} \)
29 \( 1 + (-1.53 - 4.72i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.394 - 0.286i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.28 + 10.1i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.87 - 11.9i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.45T + 43T^{2} \)
47 \( 1 + (3.10 - 9.54i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.26 - 3.09i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.76 - 8.49i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.75 - 1.27i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 0.709T + 67T^{2} \)
71 \( 1 + (-0.654 + 0.475i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.163 + 0.502i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.70 - 3.41i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.73 - 4.16i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 7.76T + 89T^{2} \)
97 \( 1 + (2.93 + 2.13i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.87002703134975084530716193911, −11.04148421601127645173985271214, −10.18193481463525114417274943673, −9.533384835446540938407287744710, −8.402838140290019784756899329686, −7.28051299086370538913348942527, −5.96518784489302888013361725591, −4.94580935050736227431203163018, −3.93993653088480038563319308027, −3.09271956520298516751249855569, 0.46373185137982069950324683347, 1.77670846439318566490327366223, 3.83607526204520398044950914289, 5.06035362756374623990051345499, 6.39432133095484835127631608804, 7.14836130885743105107687802726, 8.005702127749880342252996456946, 8.633205774956802940549630316336, 10.17820686685517034845550344991, 11.64973660600613538502195195108

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