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Properties

Label 2-392-392.29-c1-0-7
Degree $2$
Conductor $392$
Sign $-0.115 - 0.993i$
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  + (−1.35 − 0.388i)2-s + (1.66 − 0.379i)3-s + (1.69 + 1.05i)4-s + (−3.54 + 0.809i)5-s + (−2.40 − 0.129i)6-s + (0.860 + 2.50i)7-s + (−1.89 − 2.09i)8-s + (−0.0864 + 0.0416i)9-s + (5.13 + 0.277i)10-s + (1.65 − 3.44i)11-s + (3.22 + 1.11i)12-s + (−2.26 + 4.69i)13-s + (−0.197 − 3.73i)14-s + (−5.58 + 2.69i)15-s + (1.76 + 3.58i)16-s + (−4.87 + 6.11i)17-s + ⋯
L(s)  = 1  + (−0.961 − 0.274i)2-s + (0.959 − 0.218i)3-s + (0.849 + 0.528i)4-s + (−1.58 + 0.362i)5-s + (−0.982 − 0.0530i)6-s + (0.325 + 0.945i)7-s + (−0.671 − 0.741i)8-s + (−0.0288 + 0.0138i)9-s + (1.62 + 0.0876i)10-s + (0.499 − 1.03i)11-s + (0.930 + 0.320i)12-s + (−0.627 + 1.30i)13-s + (−0.0528 − 0.998i)14-s + (−1.44 + 0.694i)15-s + (0.441 + 0.897i)16-s + (−1.18 + 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.115 - 0.993i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.115 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.426961 + 0.479617i\)
\(L(\frac12)\) \(\approx\) \(0.426961 + 0.479617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.388i)T \)
7 \( 1 + (-0.860 - 2.50i)T \)
good3 \( 1 + (-1.66 + 0.379i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (3.54 - 0.809i)T + (4.50 - 2.16i)T^{2} \)
11 \( 1 + (-1.65 + 3.44i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.26 - 4.69i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (4.87 - 6.11i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + 0.772iT - 19T^{2} \)
23 \( 1 + (-2.44 - 3.06i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (2.61 + 2.08i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 - 0.677T + 31T^{2} \)
37 \( 1 + (0.607 + 0.484i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 + (-1.36 - 5.97i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (0.369 + 0.0843i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (-7.98 - 3.84i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-2.04 + 1.63i)T + (11.7 - 51.6i)T^{2} \)
59 \( 1 + (-8.57 - 1.95i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (6.62 + 5.28i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 - 5.48iT - 67T^{2} \)
71 \( 1 + (6.89 + 8.64i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-14.7 + 7.09i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 5.42T + 79T^{2} \)
83 \( 1 + (4.20 + 8.73i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (3.57 - 1.72i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 - 8.08T + 97T^{2} \)
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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

−11.41445292435784661793637913137, −10.94032383411109316116259178546, −9.243859459526549681778338587337, −8.733990463720368574041690615446, −8.125051070710044813719904641477, −7.29727728449851125026786051757, −6.23640519444885955222003544793, −4.14678230394907038882062370052, −3.16744444986371407491276919431, −2.02415992766228670246025722154, 0.49707550732844054348291560127, 2.63235474770413934320505890912, 3.93583819647635109183576518635, 4.99791248001137567928775933207, 7.11890692511783119505191157834, 7.38807467547458766267418308125, 8.323541263840964578345444041335, 9.000478028794579599062230287340, 9.961001037250469931155893384937, 10.95162513631858070363917349043

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