Properties

Label 2-1792-448.405-c0-0-0
Degree $2$
Conductor $1792$
Sign $-0.881 + 0.471i$
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  + (−0.923 − 0.382i)7-s + (−0.923 + 0.382i)9-s + (−1.08 − 0.216i)11-s + (−0.707 − 1.70i)23-s + (0.382 − 0.923i)25-s + (−1.63 + 0.324i)29-s + (−1.08 − 1.63i)37-s + (−0.382 + 1.92i)43-s + (0.707 + 0.707i)49-s + (−0.382 − 0.0761i)53-s + 63-s + (0.0761 + 0.382i)67-s + (1.30 + 0.541i)71-s + (0.923 + 0.617i)77-s + (−0.541 − 0.541i)79-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)7-s + (−0.923 + 0.382i)9-s + (−1.08 − 0.216i)11-s + (−0.707 − 1.70i)23-s + (0.382 − 0.923i)25-s + (−1.63 + 0.324i)29-s + (−1.08 − 1.63i)37-s + (−0.382 + 1.92i)43-s + (0.707 + 0.707i)49-s + (−0.382 − 0.0761i)53-s + 63-s + (0.0761 + 0.382i)67-s + (1.30 + 0.541i)71-s + (0.923 + 0.617i)77-s + (−0.541 − 0.541i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ -0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2657775540\)
\(L(\frac12)\) \(\approx\) \(0.2657775540\)
\(L(1)\) 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 \( 1 + (0.923 + 0.382i)T \)
good3 \( 1 + (0.923 - 0.382i)T^{2} \)
5 \( 1 + (-0.382 + 0.923i)T^{2} \)
11 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
13 \( 1 + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.382 + 0.923i)T^{2} \)
23 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.382 - 1.92i)T + (-0.923 - 0.382i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
59 \( 1 + (-0.382 + 0.923i)T^{2} \)
61 \( 1 + (0.923 - 0.382i)T^{2} \)
67 \( 1 + (-0.0761 - 0.382i)T + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
83 \( 1 + (0.382 + 0.923i)T^{2} \)
89 \( 1 + (0.707 + 0.707i)T^{2} \)
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

−9.096752753424612116645721966419, −8.325144572860215569189556653220, −7.66385703763298711766563866123, −6.68290948455586779289843276474, −5.93519667461127539457901259727, −5.15906729238641378254182928519, −4.08806435357031403792119358873, −3.04477361268724157054064952584, −2.28594345944014048133979888164, −0.17913457193175150975863816429, 1.97821723851981163573984527693, 3.12242116436875192285188029185, 3.68730877566549624047420401110, 5.31277456088459691968956540809, 5.57220133757209465702592509776, 6.61613060772088626910788123295, 7.44652502559755005984112836181, 8.259947959604446210890888316030, 9.130078264270227908447396215671, 9.674211213120603516957327920290

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