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Properties

Label 2-1096-1096.171-c0-0-0
Degree $2$
Conductor $1096$
Sign $-0.981 - 0.190i$
Analytic cond. $0.546975$
Root an. cond. $0.739577$
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  + (0.0922 − 0.995i)2-s + (−0.243 − 0.489i)3-s + (−0.982 − 0.183i)4-s + (−0.510 + 0.197i)6-s + (−0.273 + 0.961i)8-s + (0.422 − 0.558i)9-s + (−1.83 − 0.342i)11-s + (0.149 + 0.526i)12-s + (0.932 + 0.361i)16-s + (−0.510 − 1.79i)17-s + (−0.517 − 0.471i)18-s + (−0.404 − 0.368i)19-s + (−0.510 + 1.79i)22-s + (0.538 − 0.100i)24-s + (−0.982 + 0.183i)25-s + ⋯
L(s)  = 1  + (0.0922 − 0.995i)2-s + (−0.243 − 0.489i)3-s + (−0.982 − 0.183i)4-s + (−0.510 + 0.197i)6-s + (−0.273 + 0.961i)8-s + (0.422 − 0.558i)9-s + (−1.83 − 0.342i)11-s + (0.149 + 0.526i)12-s + (0.932 + 0.361i)16-s + (−0.510 − 1.79i)17-s + (−0.517 − 0.471i)18-s + (−0.404 − 0.368i)19-s + (−0.510 + 1.79i)22-s + (0.538 − 0.100i)24-s + (−0.982 + 0.183i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.981 - 0.190i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ -0.981 - 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6219898137\)
\(L(\frac12)\) \(\approx\) \(0.6219898137\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0922 + 0.995i)T \)
137 \( 1 + (-0.739 + 0.673i)T \)
good3 \( 1 + (0.243 + 0.489i)T + (-0.602 + 0.798i)T^{2} \)
5 \( 1 + (0.982 - 0.183i)T^{2} \)
7 \( 1 + (-0.445 + 0.895i)T^{2} \)
11 \( 1 + (1.83 + 0.342i)T + (0.932 + 0.361i)T^{2} \)
13 \( 1 + (-0.445 + 0.895i)T^{2} \)
17 \( 1 + (0.510 + 1.79i)T + (-0.850 + 0.526i)T^{2} \)
19 \( 1 + (0.404 + 0.368i)T + (0.0922 + 0.995i)T^{2} \)
23 \( 1 + (-0.739 - 0.673i)T^{2} \)
29 \( 1 + (-0.739 - 0.673i)T^{2} \)
31 \( 1 + (-0.0922 + 0.995i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.891T + T^{2} \)
43 \( 1 + (-1.37 - 1.25i)T + (0.0922 + 0.995i)T^{2} \)
47 \( 1 + (0.273 + 0.961i)T^{2} \)
53 \( 1 + (-0.0922 - 0.995i)T^{2} \)
59 \( 1 + (0.111 - 0.147i)T + (-0.273 - 0.961i)T^{2} \)
61 \( 1 + (0.273 - 0.961i)T^{2} \)
67 \( 1 + (-0.465 + 0.288i)T + (0.445 - 0.895i)T^{2} \)
71 \( 1 + (-0.932 + 0.361i)T^{2} \)
73 \( 1 + (0.757 + 0.469i)T + (0.445 + 0.895i)T^{2} \)
79 \( 1 + (0.602 + 0.798i)T^{2} \)
83 \( 1 + (-0.465 + 1.63i)T + (-0.850 - 0.526i)T^{2} \)
89 \( 1 + (-0.136 - 1.47i)T + (-0.982 + 0.183i)T^{2} \)
97 \( 1 + (-1.93 - 0.361i)T + (0.932 + 0.361i)T^{2} \)
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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

−9.690015671218948904999924217421, −9.082677219272717405184416373106, −7.952814262067867515870233732002, −7.29960072532134448465139636942, −6.04637779487629424030776490699, −5.18970591340642841394114901630, −4.31921556809610453106409284069, −3.02722227787454312454111571968, −2.22030226116450901912061679435, −0.54948366249340831418208994508, 2.22094079280064642470444996598, 3.87722718011056479422080242172, 4.54466629359101272526988183238, 5.51210352110402745537187298085, 6.06972902632981850502715064939, 7.35527890102175764572832946666, 7.891351467530308836889471801344, 8.619634317778095216113934509015, 9.723230256829359955534129384569, 10.44744723565074305932678722362

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