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Properties

Label 2-1352-104.35-c0-0-7
Degree $2$
Conductor $1352$
Sign $-0.978 + 0.207i$
Analytic cond. $0.674735$
Root an. cond. $0.821423$
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  + (0.5 − 0.866i)2-s + (0.900 − 1.56i)3-s + (−0.499 − 0.866i)4-s + (−0.900 − 1.56i)6-s − 0.999·8-s + (−1.12 − 1.94i)9-s + (−0.222 + 0.385i)11-s − 1.80·12-s + (−0.5 + 0.866i)16-s + (0.222 + 0.385i)17-s − 2.24·18-s + (0.623 + 1.07i)19-s + (0.222 + 0.385i)22-s + (−0.900 + 1.56i)24-s + 25-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.900 − 1.56i)3-s + (−0.499 − 0.866i)4-s + (−0.900 − 1.56i)6-s − 0.999·8-s + (−1.12 − 1.94i)9-s + (−0.222 + 0.385i)11-s − 1.80·12-s + (−0.5 + 0.866i)16-s + (0.222 + 0.385i)17-s − 2.24·18-s + (0.623 + 1.07i)19-s + (0.222 + 0.385i)22-s + (−0.900 + 1.56i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.978 + 0.207i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ -0.978 + 0.207i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.661863277\)
\(L(\frac12)\) \(\approx\) \(1.661863277\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 \)
good3 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - 1.80T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.24T + T^{2} \)
89 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.380754965132077124852989288220, −8.600929032073098803843577211406, −7.86059822064019796776518611522, −7.00118958751637753780633184866, −6.16174268056102106312039904857, −5.25245580340178865195701033352, −3.86747452363300173399921031713, −3.02515875906868252518361419399, −2.10301877157379025551936223004, −1.21175648757653149112021120388, 2.85347064787475089785545640890, 3.24949001757948100072023038361, 4.46233490673830805293821860899, 4.89030594789572983304321588836, 5.80172879055421406454474609590, 6.98128676215701511994825451316, 7.925594379048495896647048449588, 8.576605096705676639082313641155, 9.291578920417041778657433099042, 9.808820207824077907860162619177

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