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Properties

Label 2-1352-104.35-c0-0-6
Degree $2$
Conductor $1352$
Sign $-0.562 + 0.826i$
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 + (0.222 − 0.385i)3-s + (−0.499 − 0.866i)4-s + (−0.222 − 0.385i)6-s − 0.999·8-s + (0.400 + 0.694i)9-s + (0.623 − 1.07i)11-s − 0.445·12-s + (−0.5 + 0.866i)16-s + (−0.623 − 1.07i)17-s + 0.801·18-s + (−0.900 − 1.56i)19-s + (−0.623 − 1.07i)22-s + (−0.222 + 0.385i)24-s + 25-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.222 − 0.385i)3-s + (−0.499 − 0.866i)4-s + (−0.222 − 0.385i)6-s − 0.999·8-s + (0.400 + 0.694i)9-s + (0.623 − 1.07i)11-s − 0.445·12-s + (−0.5 + 0.866i)16-s + (−0.623 − 1.07i)17-s + 0.801·18-s + (−0.900 − 1.56i)19-s + (−0.623 − 1.07i)22-s + (−0.222 + 0.385i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.562 + 0.826i$
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: \((2,\ 1352,\ (\ :0),\ -0.562 + 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.401091465\)
\(L(\frac12)\) \(\approx\) \(1.401091465\)
\(L(1)\) 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 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.900 + 1.56i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - 0.445T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.80T + T^{2} \)
89 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.455744463768201635679034991610, −8.933613457905340327709530587786, −8.074599459792791102739107855806, −6.84213299923342862764774322351, −6.27734219424794742534284424877, −4.94592617397670818276150270801, −4.49558404607589468371002335910, −3.12857318142616945327116080614, −2.42828676032942315362870736421, −1.07585918344264337924998777782, 1.98776823714905273942920837375, 3.65439623258113674974330291893, 4.03345282107745953998135294041, 4.96978338663157786343508575529, 6.11991862271647754497003757503, 6.70690296179928431065363842414, 7.48242297981843686247408091597, 8.586680595848892228473804869820, 8.971174786440264261382495330910, 9.990388309820090580037601318949

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