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Properties

Label 2-392-56.13-c2-0-53
Degree $2$
Conductor $392$
Sign $0.598 - 0.801i$
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  + (1.75 + 0.957i)2-s + 5.56·3-s + (2.16 + 3.36i)4-s − 3.05·5-s + (9.76 + 5.32i)6-s + (0.579 + 7.97i)8-s + 21.9·9-s + (−5.36 − 2.92i)10-s + 0.122i·11-s + (12.0 + 18.7i)12-s − 4.11·13-s − 17.0·15-s + (−6.62 + 14.5i)16-s − 20.6i·17-s + (38.4 + 20.9i)18-s + 8.93·19-s + ⋯
L(s)  = 1  + (0.877 + 0.478i)2-s + 1.85·3-s + (0.541 + 0.840i)4-s − 0.611·5-s + (1.62 + 0.887i)6-s + (0.0724 + 0.997i)8-s + 2.43·9-s + (−0.536 − 0.292i)10-s + 0.0111i·11-s + (1.00 + 1.55i)12-s − 0.316·13-s − 1.13·15-s + (−0.414 + 0.910i)16-s − 1.21i·17-s + (2.13 + 1.16i)18-s + 0.470·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.598 - 0.801i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.16737 + 2.08965i\)
\(L(\frac12)\) \(\approx\) \(4.16737 + 2.08965i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.75 - 0.957i)T \)
7 \( 1 \)
good3 \( 1 - 5.56T + 9T^{2} \)
5 \( 1 + 3.05T + 25T^{2} \)
11 \( 1 - 0.122iT - 121T^{2} \)
13 \( 1 + 4.11T + 169T^{2} \)
17 \( 1 + 20.6iT - 289T^{2} \)
19 \( 1 - 8.93T + 361T^{2} \)
23 \( 1 + 15.0T + 529T^{2} \)
29 \( 1 - 31.6iT - 841T^{2} \)
31 \( 1 + 26.5iT - 961T^{2} \)
37 \( 1 - 29.0iT - 1.36e3T^{2} \)
41 \( 1 - 9.26iT - 1.68e3T^{2} \)
43 \( 1 + 45.3iT - 1.84e3T^{2} \)
47 \( 1 + 79.3iT - 2.20e3T^{2} \)
53 \( 1 + 63.5iT - 2.80e3T^{2} \)
59 \( 1 + 28.5T + 3.48e3T^{2} \)
61 \( 1 + 25.2T + 3.72e3T^{2} \)
67 \( 1 - 75.6iT - 4.48e3T^{2} \)
71 \( 1 + 2.81T + 5.04e3T^{2} \)
73 \( 1 + 12.8iT - 5.32e3T^{2} \)
79 \( 1 - 71.2T + 6.24e3T^{2} \)
83 \( 1 - 30.0T + 6.88e3T^{2} \)
89 \( 1 - 17.6iT - 7.92e3T^{2} \)
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

−11.57780888725572450569043007642, −10.10075862180543191342925009092, −9.113650717663133018030574709199, −8.208883038390976889595944995713, −7.55193924804094373241256734291, −6.84709033285413541225230006486, −5.11655269345657598793228230580, −4.00473191371044701666830772863, −3.23656206086909043721627019554, −2.17851405456194021263707774275, 1.67188463346863571319046268214, 2.82002126624358443116903141957, 3.77293095165712796782840526991, 4.46921272205163332780549507767, 6.14444573647984154025514222802, 7.46733899035925598143094268769, 8.054124521954016176611958057267, 9.234751494975012970702104316466, 9.975913204176935313445938356961, 10.95240697936104806008089356961

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