↑

Properties

Label 2-392-392.251-c1-0-41
Degree $2$
Conductor $392$
Sign $0.997 - 0.0726i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0521 + 1.41i)2-s + (1.12 − 2.33i)3-s + (−1.99 − 0.147i)4-s + (3.85 + 1.85i)5-s + (3.24 + 1.71i)6-s + (0.763 − 2.53i)7-s + (0.312 − 2.81i)8-s + (−2.32 − 2.91i)9-s + (−2.82 + 5.34i)10-s + (−0.583 + 0.732i)11-s + (−2.58 + 4.49i)12-s + (−0.397 + 0.498i)13-s + (3.53 + 1.21i)14-s + (8.66 − 6.91i)15-s + (3.95 + 0.588i)16-s + (−6.24 − 1.42i)17-s + ⋯
L(s)  = 1  + (−0.0369 + 0.999i)2-s + (0.649 − 1.34i)3-s + (−0.997 − 0.0737i)4-s + (1.72 + 0.829i)5-s + (1.32 + 0.698i)6-s + (0.288 − 0.957i)7-s + (0.110 − 0.993i)8-s + (−0.773 − 0.970i)9-s + (−0.892 + 1.69i)10-s + (−0.176 + 0.220i)11-s + (−0.747 + 1.29i)12-s + (−0.110 + 0.138i)13-s + (0.946 + 0.323i)14-s + (2.23 − 1.78i)15-s + (0.989 + 0.147i)16-s + (−1.51 − 0.345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.997 - 0.0726i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.997 - 0.0726i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89854 + 0.0690940i\)
\(L(\frac12)\) \(\approx\) \(1.89854 + 0.0690940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0521 - 1.41i)T \)
7 \( 1 + (-0.763 + 2.53i)T \)
good3 \( 1 + (-1.12 + 2.33i)T + (-1.87 - 2.34i)T^{2} \)
5 \( 1 + (-3.85 - 1.85i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (0.583 - 0.732i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.397 - 0.498i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (6.24 + 1.42i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 0.628iT - 19T^{2} \)
23 \( 1 + (-0.883 + 0.201i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (5.55 + 1.26i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 - 6.90T + 31T^{2} \)
37 \( 1 + (-7.76 - 1.77i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (3.46 - 7.18i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (1.62 - 0.781i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (3.98 - 4.99i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-4.74 + 1.08i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (6.10 + 12.6i)T + (-36.7 + 46.1i)T^{2} \)
61 \( 1 + (2.30 - 10.0i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + 6.36T + 67T^{2} \)
71 \( 1 + (5.03 - 1.15i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (2.56 - 2.04i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 6.80iT - 79T^{2} \)
83 \( 1 + (0.0185 - 0.0147i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (6.56 - 5.23i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 8.77iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.20478777105438036331571087451, −10.08110071475193964821243947095, −9.373046973530788429417212478303, −8.277224229444001548427534941890, −7.31105653542623216804996924013, −6.70728281639891287380355739294, −6.09737054720734913715474989732, −4.66035696749311485191116140946, −2.77475159448683360671119469849, −1.52119331880890036123490212643, 1.95179337975668138530262038217, 2.81552294944982464660895079936, 4.39439922871519590454311102995, 5.10248913822236651532646301826, 5.97832561914019671192426949915, 8.447899371263911688878423304371, 8.953341422802869131183715186597, 9.440011920551859143454497562250, 10.23930427026493921810552275276, 10.96696598477937688732855131298

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