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Properties

Label 2-392-56.27-c1-0-31
Degree $2$
Conductor $392$
Sign $-0.156 + 0.987i$
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  + 1.41·2-s − 3.37i·3-s + 2.00·4-s − 4.77i·6-s + 2.82·8-s − 8.41·9-s + 2.24·11-s − 6.75i·12-s + 4.00·16-s + 3.37i·17-s − 11.8·18-s + 1.39i·19-s + 3.17·22-s − 9.55i·24-s − 5·25-s + ⋯
L(s)  = 1  + 1.00·2-s − 1.95i·3-s + 1.00·4-s − 1.95i·6-s + 1.00·8-s − 2.80·9-s + 0.676·11-s − 1.95i·12-s + 1.00·16-s + 0.819i·17-s − 2.80·18-s + 0.321i·19-s + 0.676·22-s − 1.95i·24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58950 - 1.86131i\)
\(L(\frac12)\) \(\approx\) \(1.58950 - 1.86131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
7 \( 1 \)
good3 \( 1 + 3.37iT - 3T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 3.37iT - 17T^{2} \)
19 \( 1 - 1.39iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 8.15iT - 41T^{2} \)
43 \( 1 - 13.0T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 10.9iT - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 8.48T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 17.7iT - 83T^{2} \)
89 \( 1 + 12.1iT - 89T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−11.59296748127714518510175290143, −10.59040095074913453359037541923, −8.886687424290441428635575306380, −7.82093062013373173431685175971, −7.17152183556223483653360433100, −6.20494577969483761611031721415, −5.66146869234467549316520572774, −3.89467172415305145333909008026, −2.49852174616488983608155380727, −1.42251241089855163724047909662, 2.73370984822445473800108118628, 3.78752791798539252294808926140, 4.54746904003177883314268759710, 5.42634059451941202387515861568, 6.37675623192398542502351789208, 7.87610034453133981741459986903, 9.171233930762076984237980125857, 9.811952300313011638445835120686, 10.83414947562781700621008444335, 11.40433405515417077862465219235

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