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Properties

Label 2-392-392.261-c1-0-1
Degree $2$
Conductor $392$
Sign $-0.942 + 0.333i$
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  + (0.668 + 1.24i)2-s + (0.0513 − 0.00384i)3-s + (−1.10 + 1.66i)4-s + (−2.23 − 3.27i)5-s + (0.0391 + 0.0614i)6-s + (−0.442 + 2.60i)7-s + (−2.81 − 0.262i)8-s + (−2.96 + 0.446i)9-s + (2.58 − 4.97i)10-s + (−0.816 + 5.41i)11-s + (−0.0503 + 0.0898i)12-s + (−1.77 − 1.41i)13-s + (−3.54 + 1.19i)14-s + (−0.127 − 0.159i)15-s + (−1.55 − 3.68i)16-s + (−2.44 + 0.755i)17-s + ⋯
L(s)  = 1  + (0.472 + 0.881i)2-s + (0.0296 − 0.00222i)3-s + (−0.552 + 0.833i)4-s + (−0.999 − 1.46i)5-s + (0.0159 + 0.0250i)6-s + (−0.167 + 0.985i)7-s + (−0.995 − 0.0926i)8-s + (−0.987 + 0.148i)9-s + (0.818 − 1.57i)10-s + (−0.246 + 1.63i)11-s + (−0.0145 + 0.0259i)12-s + (−0.491 − 0.391i)13-s + (−0.947 + 0.318i)14-s + (−0.0328 − 0.0412i)15-s + (−0.389 − 0.921i)16-s + (−0.593 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.942 + 0.333i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0760079 - 0.442636i\)
\(L(\frac12)\) \(\approx\) \(0.0760079 - 0.442636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.668 - 1.24i)T \)
7 \( 1 + (0.442 - 2.60i)T \)
good3 \( 1 + (-0.0513 + 0.00384i)T + (2.96 - 0.447i)T^{2} \)
5 \( 1 + (2.23 + 3.27i)T + (-1.82 + 4.65i)T^{2} \)
11 \( 1 + (0.816 - 5.41i)T + (-10.5 - 3.24i)T^{2} \)
13 \( 1 + (1.77 + 1.41i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 + (2.44 - 0.755i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (2.50 + 1.44i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.39 - 1.35i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (-5.14 - 1.17i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (3.04 + 5.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.46 + 1.58i)T + (-2.76 - 36.8i)T^{2} \)
41 \( 1 + (-9.28 - 4.47i)T + (25.5 + 32.0i)T^{2} \)
43 \( 1 + (1.35 + 2.81i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-2.54 - 6.47i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (5.35 + 5.77i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (1.06 - 1.56i)T + (-21.5 - 54.9i)T^{2} \)
61 \( 1 + (9.29 - 10.0i)T + (-4.55 - 60.8i)T^{2} \)
67 \( 1 + (9.35 - 5.40i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.945 + 4.14i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.04 - 2.66i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (4.00 - 6.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.58 - 1.26i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (-1.69 + 0.255i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−12.21149232179595933303083761183, −11.29389402658544894876300394224, −9.441255393201442026469663113017, −8.866749805550117046813571603193, −8.085285289032456531947318816620, −7.28732678002096511373735432176, −5.87407922544632682721091945426, −4.93344438272722779930364400231, −4.36621613682874651984202363617, −2.68316965514972964325158703491, 0.24066332566775067713624225326, 2.83225135282819826640195226895, 3.36673233611484153179252739711, 4.46580651934995338018930649307, 6.04003025634986205819388843783, 6.84273722582029555179832856890, 8.069213524531925037768349442984, 9.064281481124888294103164647185, 10.51705653625002812539351387382, 10.89263565662892098857664408584

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