Properties

Label 2-1792-224.125-c0-0-1
Degree $2$
Conductor $1792$
Sign $0.980 + 0.195i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (0.707 − 0.292i)11-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (−0.707 − 0.292i)29-s + (0.707 + 1.70i)37-s + (1.70 − 0.707i)43-s − 1.00i·49-s + (1.70 − 0.707i)53-s + 1.00·63-s + (−1.70 − 0.707i)67-s + (0.292 − 0.707i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (0.707 − 0.292i)11-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (−0.707 − 0.292i)29-s + (0.707 + 1.70i)37-s + (1.70 − 0.707i)43-s − 1.00i·49-s + (1.70 − 0.707i)53-s + 1.00·63-s + (−1.70 − 0.707i)67-s + (0.292 − 0.707i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.322060299\)
\(L(\frac12)\) \(\approx\) \(1.322060299\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−9.536714178140544423416125788511, −8.535406472337785840626655447806, −7.83001927513730427961286916804, −7.21169908918584646357283310710, −6.30294304954170728983797977423, −5.33239588157186812095336307573, −4.34433829279644104583674958906, −3.85460609620413376151389506013, −2.32250155642156792449899178178, −1.28299193648057815971937940009, 1.41513987497040380049030504688, 2.40960796835518361618559302744, 3.86133822578618242995045437430, 4.35406246107934631704726748596, 5.64521292466912601895852583519, 6.12566235568468787622070592799, 7.30184504677746441616279397118, 7.77688573400485519565483686761, 8.946176278037502947939980122962, 9.321567562490702486880676241986

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