Properties

Label 2-1792-1792.685-c0-0-0
Degree $2$
Conductor $1792$
Sign $-0.492 + 0.870i$
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  + (−0.0980 − 0.995i)2-s + (−0.980 + 0.195i)4-s + (−0.956 − 0.290i)7-s + (0.290 + 0.956i)8-s + (−0.471 + 0.881i)9-s + (1.60 − 1.18i)11-s + (−0.195 + 0.980i)14-s + (0.923 − 0.382i)16-s + (0.923 + 0.382i)18-s + (−1.34 − 1.48i)22-s + (0.172 − 1.75i)23-s + (−0.773 − 0.634i)25-s + (0.995 + 0.0980i)28-s + (1.86 − 0.276i)29-s + (−0.471 − 0.881i)32-s + ⋯
L(s)  = 1  + (−0.0980 − 0.995i)2-s + (−0.980 + 0.195i)4-s + (−0.956 − 0.290i)7-s + (0.290 + 0.956i)8-s + (−0.471 + 0.881i)9-s + (1.60 − 1.18i)11-s + (−0.195 + 0.980i)14-s + (0.923 − 0.382i)16-s + (0.923 + 0.382i)18-s + (−1.34 − 1.48i)22-s + (0.172 − 1.75i)23-s + (−0.773 − 0.634i)25-s + (0.995 + 0.0980i)28-s + (1.86 − 0.276i)29-s + (−0.471 − 0.881i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.492 + 0.870i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ -0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8413682031\)
\(L(\frac12)\) \(\approx\) \(0.8413682031\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0980 + 0.995i)T \)
7 \( 1 + (0.956 + 0.290i)T \)
good3 \( 1 + (0.471 - 0.881i)T^{2} \)
5 \( 1 + (0.773 + 0.634i)T^{2} \)
11 \( 1 + (-1.60 + 1.18i)T + (0.290 - 0.956i)T^{2} \)
13 \( 1 + (0.634 + 0.773i)T^{2} \)
17 \( 1 + (-0.923 - 0.382i)T^{2} \)
19 \( 1 + (-0.995 - 0.0980i)T^{2} \)
23 \( 1 + (-0.172 + 1.75i)T + (-0.980 - 0.195i)T^{2} \)
29 \( 1 + (-1.86 + 0.276i)T + (0.956 - 0.290i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (1.19 + 1.07i)T + (0.0980 + 0.995i)T^{2} \)
41 \( 1 + (-0.195 + 0.980i)T^{2} \)
43 \( 1 + (-0.251 - 0.150i)T + (0.471 + 0.881i)T^{2} \)
47 \( 1 + (-0.382 + 0.923i)T^{2} \)
53 \( 1 + (1.91 + 0.284i)T + (0.956 + 0.290i)T^{2} \)
59 \( 1 + (0.634 - 0.773i)T^{2} \)
61 \( 1 + (0.881 + 0.471i)T^{2} \)
67 \( 1 + (-1.66 - 0.416i)T + (0.881 + 0.471i)T^{2} \)
71 \( 1 + (-0.979 + 0.523i)T + (0.555 - 0.831i)T^{2} \)
73 \( 1 + (-0.831 + 0.555i)T^{2} \)
79 \( 1 + (-0.704 + 1.05i)T + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (-0.0980 + 0.995i)T^{2} \)
89 \( 1 + (0.980 - 0.195i)T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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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

−9.224268383021230360573622234080, −8.627309757263577889007850171504, −7.998637749172163710999126525758, −6.62496103902310166294797842204, −6.07981262843540501841544541105, −4.87290679788032821305011411818, −3.97765510597344670230435339605, −3.20249703442799407465133187577, −2.25982378835590047994265147658, −0.73778193474072524286579131944, 1.39707393629494513039505177592, 3.28647646727335537603935918508, 3.91890162886588906355905593084, 5.01249558475049120049446949798, 5.96174492544780660713238828984, 6.69494432368820500671665914249, 7.00970757711217610131590695389, 8.184603161840023964815383154421, 9.033928855005327442564150563189, 9.632383765150652051964745191912

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