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Properties

Label 2-352-11.4-c1-0-8
Degree $2$
Conductor $352$
Sign $-0.751 + 0.659i$
Analytic cond. $2.81073$
Root an. cond. $1.67652$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 1.53i)3-s + (−2.61 + 1.90i)5-s + (−1.38 − 4.25i)7-s + (0.309 + 0.224i)9-s + (−3.30 + 0.224i)11-s + (−1 − 0.726i)13-s + (−1.61 − 4.97i)15-s + (−1.5 + 1.08i)17-s + (1.66 − 5.11i)19-s + 7.23·21-s − 5.23·23-s + (1.69 − 5.20i)25-s + (−4.42 + 3.21i)27-s + (−2.61 − 8.05i)29-s + (0.381 + 0.277i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.888i)3-s + (−1.17 + 0.850i)5-s + (−0.522 − 1.60i)7-s + (0.103 + 0.0748i)9-s + (−0.997 + 0.0676i)11-s + (−0.277 − 0.201i)13-s + (−0.417 − 1.28i)15-s + (−0.363 + 0.264i)17-s + (0.381 − 1.17i)19-s + 1.57·21-s − 1.09·23-s + (0.338 − 1.04i)25-s + (−0.851 + 0.619i)27-s + (−0.486 − 1.49i)29-s + (0.0686 + 0.0498i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $-0.751 + 0.659i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ -0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 - 0.224i)T \)
good3 \( 1 + (0.5 - 1.53i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.61 - 1.90i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.38 + 4.25i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1 + 0.726i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.5 - 1.08i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.66 + 5.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.23T + 23T^{2} \)
29 \( 1 + (2.61 + 8.05i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.381 - 0.277i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.85 - 8.78i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.26 - 6.96i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.09T + 43T^{2} \)
47 \( 1 + (2 - 6.15i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.23 + 3.80i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.80 - 8.64i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2 - 1.45i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.38T + 67T^{2} \)
71 \( 1 + (-3 + 2.17i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.26 + 3.88i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.85 + 3.52i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.92 - 1.40i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 6.38T + 89T^{2} \)
97 \( 1 + (4.54 + 3.30i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.02057187369932880474590405706, −10.23437906467793589238101379297, −9.790307239538750785277506975410, −7.946437598492409176256447020654, −7.46098195988438707761063349612, −6.45311900967382577253584636680, −4.74593644084615635962368525140, −4.06661752714080717231614769577, −3.04719940887095986562796251707, 0, 2.04981031053179698684337632579, 3.64199558518092546762566709298, 5.13256061666897703691362413305, 5.96127071999746570565225057537, 7.22494039507714965586468935351, 8.066893630161135541193811681531, 8.833391827813415079616245940941, 9.860766434134297041671188187853, 11.31890009626581160904711270912

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