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Properties

Label 2-392-7.2-c3-0-14
Degree $2$
Conductor $392$
Sign $0.701 + 0.712i$
Analytic cond. $23.1287$
Root an. cond. $4.80923$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 3.46i)3-s + (−1 + 1.73i)5-s + (5.50 − 9.52i)9-s + (22 + 38.1i)11-s − 22·13-s + 7.99·15-s + (25 + 43.3i)17-s + (22 − 38.1i)19-s + (28 − 48.4i)23-s + (60.5 + 104. i)25-s − 152·27-s + 198·29-s + (−80 − 138. i)31-s + (88 − 152. i)33-s + (81 − 140. i)37-s + ⋯
L(s)  = 1  + (−0.384 − 0.666i)3-s + (−0.0894 + 0.154i)5-s + (0.203 − 0.352i)9-s + (0.603 + 1.04i)11-s − 0.469·13-s + 0.137·15-s + (0.356 + 0.617i)17-s + (0.265 − 0.460i)19-s + (0.253 − 0.439i)23-s + (0.483 + 0.838i)25-s − 1.08·27-s + 1.26·29-s + (−0.463 − 0.802i)31-s + (0.464 − 0.804i)33-s + (0.359 − 0.623i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(23.1287\)
Root analytic conductor: \(4.80923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :3/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.666481511\)
\(L(\frac12)\) \(\approx\) \(1.666481511\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (2 + 3.46i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-22 - 38.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 22T + 2.19e3T^{2} \)
17 \( 1 + (-25 - 43.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-22 + 38.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-28 + 48.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 198T + 2.43e4T^{2} \)
31 \( 1 + (80 + 138. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-81 + 140. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 198T + 6.89e4T^{2} \)
43 \( 1 - 52T + 7.95e4T^{2} \)
47 \( 1 + (-264 + 457. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-121 - 209. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (334 + 578. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-275 + 476. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (94 + 162. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 728T + 3.57e5T^{2} \)
73 \( 1 + (-77 - 133. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-328 + 568. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 236T + 5.71e5T^{2} \)
89 \( 1 + (-357 + 618. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 478T + 9.12e5T^{2} \)
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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

−10.86836962616143423273641604569, −9.813771470640864068937084135797, −9.047546597725363398469666008409, −7.68156320020486628582263507665, −6.99538993103111590910572603500, −6.19454444102904382143562379728, −4.91404239474322242323797458641, −3.73553763954389711430663425681, −2.14052824341776905060004784721, −0.803415081825584226609011280846, 0.996109704592799223037241894682, 2.87911282184555947557801438490, 4.15752629804938100128320112584, 5.09087468445619245484773506076, 6.05067649402757388551715655961, 7.26081620232305120521733322607, 8.309484312490223074343821163216, 9.287826801813277844128368181719, 10.15986095689475965479479086140, 10.94302571076621442534087672592

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