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Properties

Label 2-392-56.45-c2-0-74
Degree $2$
Conductor $392$
Sign $-0.507 - 0.861i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0483 − 1.99i)2-s + (2.78 − 4.81i)3-s + (−3.99 + 0.193i)4-s + (−1.52 − 2.64i)5-s + (−9.76 − 5.32i)6-s + (0.579 + 7.97i)8-s + (−10.9 − 18.9i)9-s + (−5.22 + 3.18i)10-s + (−0.106 − 0.0612i)11-s + (−10.1 + 19.7i)12-s + 4.11·13-s − 17.0·15-s + (15.9 − 1.54i)16-s + (−17.8 − 10.3i)17-s + (−37.4 + 22.8i)18-s + (4.46 + 7.74i)19-s + ⋯
L(s)  = 1  + (−0.0241 − 0.999i)2-s + (0.926 − 1.60i)3-s + (−0.998 + 0.0483i)4-s + (−0.305 − 0.529i)5-s + (−1.62 − 0.887i)6-s + (0.0724 + 0.997i)8-s + (−1.21 − 2.10i)9-s + (−0.522 + 0.318i)10-s + (−0.00963 − 0.00556i)11-s + (−0.848 + 1.64i)12-s + 0.316·13-s − 1.13·15-s + (0.995 − 0.0965i)16-s + (−1.05 − 0.606i)17-s + (−2.07 + 1.26i)18-s + (0.235 + 0.407i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.507 - 0.861i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.507 - 0.861i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.753726 + 1.31937i\)
\(L(\frac12)\) \(\approx\) \(0.753726 + 1.31937i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0483 + 1.99i)T \)
7 \( 1 \)
good3 \( 1 + (-2.78 + 4.81i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.52 + 2.64i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (0.106 + 0.0612i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 4.11T + 169T^{2} \)
17 \( 1 + (17.8 + 10.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-4.46 - 7.74i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-7.51 - 13.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 31.6iT - 841T^{2} \)
31 \( 1 + (23.0 + 13.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-25.1 + 14.5i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 9.26iT - 1.68e3T^{2} \)
43 \( 1 + 45.3iT - 1.84e3T^{2} \)
47 \( 1 + (-68.6 + 39.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-55.0 - 31.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (14.2 - 24.6i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (12.6 + 21.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (65.4 + 37.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 2.81T + 5.04e3T^{2} \)
73 \( 1 + (11.0 + 6.40i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (35.6 + 61.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 30.0T + 6.88e3T^{2} \)
89 \( 1 + (15.3 - 8.83i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 26.1iT - 9.40e3T^{2} \)
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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

−10.67488581473012410090991985153, −9.104198809196338576657957034199, −8.867090345194965350674992531847, −7.84412036516334857836876953510, −7.00825962831666350503616626275, −5.59310895241987758686489132498, −4.04988113163076249630698937908, −2.87248942329802000514305976553, −1.80152812118618761565463945877, −0.61307445676399718254825407048, 2.86672782621807151465663062243, 3.98607582403184444561914659203, 4.65526538626944906859961652283, 5.87827539067706301513367142839, 7.15548238954650757082843652382, 8.214211840400405133980854327452, 8.892685254511423635332152029735, 9.624011651467209207689631106611, 10.55544264581297301240057884703, 11.24578379089045304989376744474

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