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Properties

Label 2-352-88.21-c4-0-24
Degree $2$
Conductor $352$
Sign $1$
Analytic cond. $36.3862$
Root an. cond. $6.03209$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81·9-s + 121·11-s + 14·13-s − 686·19-s − 706·23-s + 625·25-s + 1.48e3·29-s + 1.24e3·31-s − 1.93e3·43-s + 4.38e3·47-s + 2.40e3·49-s + 1.35e3·61-s + 7.16e3·71-s + 6.56e3·81-s + 1.10e3·83-s + 1.44e4·89-s + 6.14e3·97-s + 9.80e3·99-s − 1.72e4·101-s + 7.29e3·103-s + 1.02e4·107-s − 6.51e3·109-s − 9.66e3·113-s + 1.13e3·117-s + ⋯
L(s)  = 1  + 9-s + 11-s + 0.0828·13-s − 1.90·19-s − 1.33·23-s + 25-s + 1.76·29-s + 1.29·31-s − 1.04·43-s + 1.98·47-s + 49-s + 0.364·61-s + 1.42·71-s + 81-s + 0.160·83-s + 1.82·89-s + 0.653·97-s + 99-s − 1.68·101-s + 0.687·103-s + 0.893·107-s − 0.548·109-s − 0.756·113-s + 0.0828·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $1$
Analytic conductor: \(36.3862\)
Root analytic conductor: \(6.03209\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: $\chi_{352} (241, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.270331321\)
\(L(\frac12)\) \(\approx\) \(2.270331321\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - p^{2} T \)
good3 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
5 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
7 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
13 \( 1 - 14 T + p^{4} T^{2} \)
17 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
19 \( 1 + 686 T + p^{4} T^{2} \)
23 \( 1 + 706 T + p^{4} T^{2} \)
29 \( 1 - 1486 T + p^{4} T^{2} \)
31 \( 1 - 1246 T + p^{4} T^{2} \)
37 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
41 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
43 \( 1 + 1934 T + p^{4} T^{2} \)
47 \( 1 - 4382 T + p^{4} T^{2} \)
53 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( 1 - 1358 T + p^{4} T^{2} \)
67 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
71 \( 1 - 7166 T + p^{4} T^{2} \)
73 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
79 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
83 \( 1 - 1106 T + p^{4} T^{2} \)
89 \( 1 - 14434 T + p^{4} T^{2} \)
97 \( 1 - 6146 T + p^{4} T^{2} \)
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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

−10.63209689224837246132025072673, −10.09315435544579793182047432783, −8.927983682959890518817204432411, −8.145562697067321023903942918798, −6.81468940014758927589487257916, −6.27599470648129521883367569126, −4.64268771895569790965089441311, −3.92677717808513430689150460198, −2.27665580111374130682338968002, −0.943654841119074918044696358531, 0.943654841119074918044696358531, 2.27665580111374130682338968002, 3.92677717808513430689150460198, 4.64268771895569790965089441311, 6.27599470648129521883367569126, 6.81468940014758927589487257916, 8.145562697067321023903942918798, 8.927983682959890518817204432411, 10.09315435544579793182047432783, 10.63209689224837246132025072673

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