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Properties

Label 2-392-8.5-c1-0-35
Degree $2$
Conductor $392$
Sign $-0.634 - 0.773i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
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  + (−0.321 − 1.37i)2-s − 2.86i·3-s + (−1.79 + 0.884i)4-s − 3.19i·5-s + (−3.95 + 0.920i)6-s + (1.79 + 2.18i)8-s − 5.22·9-s + (−4.40 + 1.02i)10-s − 1.45i·11-s + (2.53 + 5.14i)12-s + 1.14i·13-s − 9.17·15-s + (2.43 − 3.17i)16-s + 5.58·17-s + (1.67 + 7.20i)18-s − 1.47i·19-s + ⋯
L(s)  = 1  + (−0.227 − 0.973i)2-s − 1.65i·3-s + (−0.896 + 0.442i)4-s − 1.43i·5-s + (−1.61 + 0.375i)6-s + (0.634 + 0.773i)8-s − 1.74·9-s + (−1.39 + 0.324i)10-s − 0.437i·11-s + (0.732 + 1.48i)12-s + 0.317i·13-s − 2.36·15-s + (0.608 − 0.793i)16-s + 1.35·17-s + (0.395 + 1.69i)18-s − 0.338i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.634 - 0.773i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.634 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435743 + 0.921048i\)
\(L(\frac12)\) \(\approx\) \(0.435743 + 0.921048i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.321 + 1.37i)T \)
7 \( 1 \)
good3 \( 1 + 2.86iT - 3T^{2} \)
5 \( 1 + 3.19iT - 5T^{2} \)
11 \( 1 + 1.45iT - 11T^{2} \)
13 \( 1 - 1.14iT - 13T^{2} \)
17 \( 1 - 5.58T + 17T^{2} \)
19 \( 1 + 1.47iT - 19T^{2} \)
23 \( 1 - 3.28T + 23T^{2} \)
29 \( 1 - 3.59iT - 29T^{2} \)
31 \( 1 + 1.01T + 31T^{2} \)
37 \( 1 - 6.49iT - 37T^{2} \)
41 \( 1 - 7.39T + 41T^{2} \)
43 \( 1 + 7.59iT - 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 + 6.31iT - 59T^{2} \)
61 \( 1 + 3.19iT - 61T^{2} \)
67 \( 1 + 2.60iT - 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 9.21T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 + 7.87iT - 83T^{2} \)
89 \( 1 - 3.23T + 89T^{2} \)
97 \( 1 - 0.401T + 97T^{2} \)
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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

−11.06151008234251349361681212917, −9.664051895581137304424692049806, −8.740960120365048761881344849754, −8.164571729022533759222689880994, −7.24096309194957368107024926400, −5.78588583184616329229243960555, −4.79045812014291564448555449879, −3.15416746697275010848873569171, −1.65160053401179721235744196600, −0.800064551783334687029690780270, 3.10583081429495138267393475388, 4.04823200105520484027635334821, 5.21339696227003682301634057638, 6.08467810220135119915781936818, 7.23668626922865542670876714969, 8.152100721658622025016458747411, 9.427197062492909991582474872255, 9.967068277222225050720147397325, 10.58636124233510060028982893383, 11.42586140192968838906682400493

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