Properties

Label 2-392-7.4-c1-0-8
Degree $2$
Conductor $392$
Sign $-0.386 + 0.922i$
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  + (1.5 − 2.59i)3-s + (−0.5 − 0.866i)5-s + (−3 − 5.19i)9-s + (0.5 − 0.866i)11-s − 2·13-s − 3·15-s + (1.5 − 2.59i)17-s + (2.5 + 4.33i)19-s + (1.5 + 2.59i)23-s + (2 − 3.46i)25-s − 9·27-s − 6·29-s + (−0.5 + 0.866i)31-s + (−1.5 − 2.59i)33-s + (2.5 + 4.33i)37-s + ⋯
L(s)  = 1  + (0.866 − 1.49i)3-s + (−0.223 − 0.387i)5-s + (−1 − 1.73i)9-s + (0.150 − 0.261i)11-s − 0.554·13-s − 0.774·15-s + (0.363 − 0.630i)17-s + (0.573 + 0.993i)19-s + (0.312 + 0.541i)23-s + (0.400 − 0.692i)25-s − 1.73·27-s − 1.11·29-s + (−0.0898 + 0.155i)31-s + (−0.261 − 0.452i)33-s + (0.410 + 0.711i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.913334 - 1.37305i\)
\(L(\frac12)\) \(\approx\) \(0.913334 - 1.37305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−11.38297036070779748274701135141, −9.837606553933393374078600441490, −8.982878980813559126953368253738, −8.026412520184841417417276467971, −7.48948416667953569758507844841, −6.52002851831898887759155111537, −5.32816108463016141071664116140, −3.64646895098528413727108001516, −2.45972080277294128346330304358, −1.07786621190767853765166755996, 2.53994449690370517419655343476, 3.56368661710612302985632906631, 4.49896427663685018480336596383, 5.50990388671335941277239073890, 7.10965975971636938455104081373, 8.041321280964769562750285933451, 9.190844393729591094545025212868, 9.557369893810969516249526129355, 10.67741216893532227023482080620, 11.14843981514927834601121037548

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