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Properties

Label 2-392-8.5-c1-0-23
Degree $2$
Conductor $392$
Sign $-0.830 + 0.557i$
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  + (−0.933 − 1.06i)2-s − 1.57i·3-s + (−0.258 + 1.98i)4-s + 0.549i·5-s + (−1.67 + 1.47i)6-s + (2.34 − 1.57i)8-s + 0.517·9-s + (0.584 − 0.512i)10-s − 2.39i·11-s + (3.12 + 0.407i)12-s − 3.96i·13-s + 0.866·15-s + (−3.86 − 1.02i)16-s − 4.21·17-s + (−0.482 − 0.549i)18-s − 6.64i·19-s + ⋯
L(s)  = 1  + (−0.659 − 0.751i)2-s − 0.909i·3-s + (−0.129 + 0.991i)4-s + 0.245i·5-s + (−0.683 + 0.600i)6-s + (0.830 − 0.557i)8-s + 0.172·9-s + (0.184 − 0.162i)10-s − 0.720i·11-s + (0.902 + 0.117i)12-s − 1.10i·13-s + 0.223·15-s + (−0.966 − 0.256i)16-s − 1.02·17-s + (−0.113 − 0.129i)18-s − 1.52i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.830 + 0.557i$
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: \((2,\ 392,\ (\ :1/2),\ -0.830 + 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252794 - 0.830637i\)
\(L(\frac12)\) \(\approx\) \(0.252794 - 0.830637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.933 + 1.06i)T \)
7 \( 1 \)
good3 \( 1 + 1.57iT - 3T^{2} \)
5 \( 1 - 0.549iT - 5T^{2} \)
11 \( 1 + 2.39iT - 11T^{2} \)
13 \( 1 + 3.96iT - 13T^{2} \)
17 \( 1 + 4.21T + 17T^{2} \)
19 \( 1 + 6.64iT - 19T^{2} \)
23 \( 1 + 2.34T + 23T^{2} \)
29 \( 1 - 8.21iT - 29T^{2} \)
31 \( 1 - 0.866T + 31T^{2} \)
37 \( 1 - 0.265iT - 37T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 5.35iT - 43T^{2} \)
47 \( 1 - 2.59T + 47T^{2} \)
53 \( 1 + 10.8iT - 53T^{2} \)
59 \( 1 + 3.77iT - 59T^{2} \)
61 \( 1 - 7.13iT - 61T^{2} \)
67 \( 1 - 2.67iT - 67T^{2} \)
71 \( 1 - 8.76T + 71T^{2} \)
73 \( 1 - 4.66T + 73T^{2} \)
79 \( 1 - 0.616T + 79T^{2} \)
83 \( 1 + 1.09iT - 83T^{2} \)
89 \( 1 + 6.39T + 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−10.91605431055401573089102646510, −10.24139998793753997008192010662, −8.964128485713836938781693749547, −8.318081441954289681484823952617, −7.22724265461325562190675185896, −6.61061835780468789962755124691, −4.93923382119211629258782095022, −3.38790397842800590323355013628, −2.28301828672501100161258533487, −0.73523514972831011725535412784, 1.84183388713679076777274024460, 4.14783250455454343742991839441, 4.75433086410316387984460278405, 6.07281327645144160197619587240, 7.01105196432241316608489987395, 8.079057393120512090761618573638, 9.071274822713847240475298912526, 9.737178687559601385722584573574, 10.39203857649529811841837914425, 11.37469187319988892501798713940

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