Properties

Label 2-392-392.299-c1-0-23
Degree $2$
Conductor $392$
Sign $0.966 + 0.255i$
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  + (0.471 − 1.33i)2-s + (0.423 − 0.620i)3-s + (−1.55 − 1.25i)4-s + (−0.325 + 4.33i)5-s + (−0.628 − 0.857i)6-s + (1.83 + 1.90i)7-s + (−2.41 + 1.47i)8-s + (0.889 + 2.26i)9-s + (5.63 + 2.48i)10-s + (2.18 − 5.56i)11-s + (−1.43 + 0.432i)12-s + (−0.323 − 0.405i)13-s + (3.40 − 1.54i)14-s + (2.55 + 2.03i)15-s + (0.834 + 3.91i)16-s + (2.75 + 2.96i)17-s + ⋯
L(s)  = 1  + (0.333 − 0.942i)2-s + (0.244 − 0.358i)3-s + (−0.777 − 0.629i)4-s + (−0.145 + 1.94i)5-s + (−0.256 − 0.350i)6-s + (0.693 + 0.720i)7-s + (−0.852 + 0.522i)8-s + (0.296 + 0.755i)9-s + (1.78 + 0.784i)10-s + (0.658 − 1.67i)11-s + (−0.415 + 0.124i)12-s + (−0.0897 − 0.112i)13-s + (0.910 − 0.412i)14-s + (0.660 + 0.526i)15-s + (0.208 + 0.978i)16-s + (0.667 + 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.966 + 0.255i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.966 + 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67764 - 0.217981i\)
\(L(\frac12)\) \(\approx\) \(1.67764 - 0.217981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.471 + 1.33i)T \)
7 \( 1 + (-1.83 - 1.90i)T \)
good3 \( 1 + (-0.423 + 0.620i)T + (-1.09 - 2.79i)T^{2} \)
5 \( 1 + (0.325 - 4.33i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (-2.18 + 5.56i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (0.323 + 0.405i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-2.75 - 2.96i)T + (-1.27 + 16.9i)T^{2} \)
19 \( 1 + (-1.66 - 0.959i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.06 - 2.22i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (-1.73 + 0.396i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (1.89 + 3.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.10 + 6.83i)T + (-30.5 - 20.8i)T^{2} \)
41 \( 1 + (-1.51 - 3.15i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (4.68 + 2.25i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (4.17 - 0.629i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (2.12 + 6.89i)T + (-43.7 + 29.8i)T^{2} \)
59 \( 1 + (-6.77 + 0.507i)T + (58.3 - 8.79i)T^{2} \)
61 \( 1 + (4.43 + 1.36i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (-1.30 - 2.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.5 - 2.63i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.342 + 2.27i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 + (4.06 + 2.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.9 + 8.73i)T + (18.4 + 80.9i)T^{2} \)
89 \( 1 + (-12.5 + 4.93i)T + (65.2 - 60.5i)T^{2} \)
97 \( 1 + 9.98iT - 97T^{2} \)
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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

−11.30056449426446406498278062505, −10.66601515846320872326397985112, −9.785875014013774251341168285039, −8.440805547131915428124514860474, −7.70591316398951313994754998123, −6.29556059631260168811695303196, −5.55085033954073764944385776106, −3.79504732419718099259311419292, −2.96011580984783027112341937197, −1.87332879032362067202115868505, 1.16583507876998381623576463293, 3.90110128667399257318178823483, 4.60256897251692660565947237996, 5.13131320018011808013991801861, 6.72989511875843591395096855112, 7.66303225842007371250221702127, 8.501869236529697801376463079883, 9.436667442829122351571513921436, 9.831389329074529326423087497365, 11.95413601571936377759043765834

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