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Properties

Label 2-392-8.5-c1-0-30
Degree $2$
Conductor $392$
Sign $0.493 + 0.869i$
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  + (1.32 + 0.486i)2-s − 2.45i·3-s + (1.52 + 1.29i)4-s − 1.48i·5-s + (1.19 − 3.26i)6-s + (1.39 + 2.45i)8-s − 3.05·9-s + (0.723 − 1.97i)10-s − 5.04i·11-s + (3.18 − 3.75i)12-s + 2.58i·13-s − 3.65·15-s + (0.655 + 3.94i)16-s − 1.25·17-s + (−4.05 − 1.48i)18-s + 3.09i·19-s + ⋯
L(s)  = 1  + (0.938 + 0.344i)2-s − 1.42i·3-s + (0.762 + 0.646i)4-s − 0.664i·5-s + (0.489 − 1.33i)6-s + (0.493 + 0.869i)8-s − 1.01·9-s + (0.228 − 0.623i)10-s − 1.52i·11-s + (0.918 − 1.08i)12-s + 0.717i·13-s − 0.943·15-s + (0.163 + 0.986i)16-s − 0.305·17-s + (−0.954 − 0.350i)18-s + 0.710i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.493 + 0.869i$
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: \((2,\ 392,\ (\ :1/2),\ 0.493 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02631 - 1.17997i\)
\(L(\frac12)\) \(\approx\) \(2.02631 - 1.17997i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 - 0.486i)T \)
7 \( 1 \)
good3 \( 1 + 2.45iT - 3T^{2} \)
5 \( 1 + 1.48iT - 5T^{2} \)
11 \( 1 + 5.04iT - 11T^{2} \)
13 \( 1 - 2.58iT - 13T^{2} \)
17 \( 1 + 1.25T + 17T^{2} \)
19 \( 1 - 3.09iT - 19T^{2} \)
23 \( 1 + 1.39T + 23T^{2} \)
29 \( 1 - 0.638iT - 29T^{2} \)
31 \( 1 - 3.65T + 31T^{2} \)
37 \( 1 - 6.02iT - 37T^{2} \)
41 \( 1 + 6.36T + 41T^{2} \)
43 \( 1 + 1.02iT - 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 5.76iT - 53T^{2} \)
59 \( 1 - 3.48iT - 59T^{2} \)
61 \( 1 - 12.8iT - 61T^{2} \)
67 \( 1 - 0.512iT - 67T^{2} \)
71 \( 1 + 7.41T + 71T^{2} \)
73 \( 1 + 9.89T + 73T^{2} \)
79 \( 1 - 8.70T + 79T^{2} \)
83 \( 1 - 2.97iT - 83T^{2} \)
89 \( 1 - 2.58T + 89T^{2} \)
97 \( 1 - 1.57T + 97T^{2} \)
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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

−11.76137148340882761488070724501, −10.60155837923363200000722818598, −8.775188950187105848701650058102, −8.223767791660381380243316847060, −7.19802103486114021231377409716, −6.32863940787586411390951586184, −5.60132708062993998315523282054, −4.25169144473771703948476097235, −2.83100155948922571414157172235, −1.37348410932916969372642022898, 2.40509341114132331186543563588, 3.53540970760200223773089751692, 4.52943697197997362586568992524, 5.21101328408868354752585994667, 6.52593117647984732282894043195, 7.47841761013451412174644777300, 9.135621045758206130881322058970, 10.07141562856129680996871339823, 10.48525068912512725997642273444, 11.29846061458204018144741402058

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