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Properties

Label 2-392-56.13-c2-0-18
Degree $2$
Conductor $392$
Sign $0.932 - 0.359i$
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  + (0.324 − 1.97i)2-s + 0.253·3-s + (−3.78 − 1.28i)4-s − 3.57·5-s + (0.0821 − 0.499i)6-s + (−3.75 + 7.06i)8-s − 8.93·9-s + (−1.15 + 7.04i)10-s + 7.88i·11-s + (−0.959 − 0.324i)12-s + 18.1·13-s − 0.904·15-s + (12.7 + 9.70i)16-s + 9.53i·17-s + (−2.89 + 17.6i)18-s + 24.8·19-s + ⋯
L(s)  = 1  + (0.162 − 0.986i)2-s + 0.0844·3-s + (−0.947 − 0.320i)4-s − 0.714·5-s + (0.0136 − 0.0833i)6-s + (−0.469 + 0.882i)8-s − 0.992·9-s + (−0.115 + 0.704i)10-s + 0.716i·11-s + (−0.0799 − 0.0270i)12-s + 1.39·13-s − 0.0603·15-s + (0.794 + 0.606i)16-s + 0.561i·17-s + (−0.161 + 0.979i)18-s + 1.30·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.359i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.932 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.932 - 0.359i$
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: \((2,\ 392,\ (\ :1),\ 0.932 - 0.359i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.00267 + 0.186686i\)
\(L(\frac12)\) \(\approx\) \(1.00267 + 0.186686i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.324 + 1.97i)T \)
7 \( 1 \)
good3 \( 1 - 0.253T + 9T^{2} \)
5 \( 1 + 3.57T + 25T^{2} \)
11 \( 1 - 7.88iT - 121T^{2} \)
13 \( 1 - 18.1T + 169T^{2} \)
17 \( 1 - 9.53iT - 289T^{2} \)
19 \( 1 - 24.8T + 361T^{2} \)
23 \( 1 + 4.29T + 529T^{2} \)
29 \( 1 - 28.3iT - 841T^{2} \)
31 \( 1 - 32.6iT - 961T^{2} \)
37 \( 1 - 29.9iT - 1.36e3T^{2} \)
41 \( 1 + 45.2iT - 1.68e3T^{2} \)
43 \( 1 - 24.9iT - 1.84e3T^{2} \)
47 \( 1 - 50.8iT - 2.20e3T^{2} \)
53 \( 1 - 62.8iT - 2.80e3T^{2} \)
59 \( 1 + 74.0T + 3.48e3T^{2} \)
61 \( 1 - 50.5T + 3.72e3T^{2} \)
67 \( 1 + 125. iT - 4.48e3T^{2} \)
71 \( 1 + 5.33T + 5.04e3T^{2} \)
73 \( 1 - 27.3iT - 5.32e3T^{2} \)
79 \( 1 + 103.T + 6.24e3T^{2} \)
83 \( 1 + 51.5T + 6.88e3T^{2} \)
89 \( 1 - 153. iT - 7.92e3T^{2} \)
97 \( 1 - 47.0iT - 9.40e3T^{2} \)
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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

−11.20481268391367803180422525616, −10.51009151253605133646297769338, −9.337139256642701103793455583950, −8.567942482799929276341107361257, −7.70325184056892729188228316540, −6.15167527624288630433104132097, −5.08231205243449425398871238831, −3.85037098110218500083385756531, −3.03962937434238168229897612147, −1.37103971733689724955895776766, 0.46899606368211948045333524669, 3.18655204300923628628110018944, 4.04536372901520336759356234973, 5.49807316430682507254707285648, 6.11979693782785247026558846547, 7.40620487532478851957881653878, 8.201800015715575101245012049243, 8.828914203990875574776170226876, 9.877036287276514420522037917115, 11.44632502278197616103992216698

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