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Properties

Label 2-1352-104.75-c0-0-7
Degree $2$
Conductor $1352$
Sign $-0.654 + 0.756i$
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  + (0.866 − 0.5i)2-s + (−0.623 − 1.07i)3-s + (0.499 − 0.866i)4-s + (−1.07 − 0.623i)6-s − 0.999i·8-s + (−0.277 + 0.480i)9-s + (1.56 − 0.900i)11-s − 1.24·12-s + (−0.5 − 0.866i)16-s + (−0.900 + 1.56i)17-s + 0.554i·18-s + (0.385 + 0.222i)19-s + (0.900 − 1.56i)22-s + (−1.07 + 0.623i)24-s − 25-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.623 − 1.07i)3-s + (0.499 − 0.866i)4-s + (−1.07 − 0.623i)6-s − 0.999i·8-s + (−0.277 + 0.480i)9-s + (1.56 − 0.900i)11-s − 1.24·12-s + (−0.5 − 0.866i)16-s + (−0.900 + 1.56i)17-s + 0.554i·18-s + (0.385 + 0.222i)19-s + (0.900 − 1.56i)22-s + (−1.07 + 0.623i)24-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.654 + 0.756i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ -0.654 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.508837110\)
\(L(\frac12)\) \(\approx\) \(1.508837110\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 \)
good3 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.56 + 0.900i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.07 + 0.623i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + 1.24iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.445iT - T^{2} \)
89 \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.56 - 0.900i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.626244091205069871328497249054, −8.729045054511214984126468325252, −7.64653695793523090476586364911, −6.54070925565091418974200912784, −6.31105044219775676189606161963, −5.55287443994028821207990106508, −4.19510037635468036946344109832, −3.53990925912837626199433500533, −2.00361569200595320511465667516, −1.15845048052267450051067574720, 2.17439185526458603448798311781, 3.61590078808975093822178733759, 4.30158432407747337683352836632, 4.95269528242381955095193088880, 5.73239152788725184673546413565, 6.82207459152227244022637196214, 7.21294649349698783615485804804, 8.549807317733674692153227293108, 9.436139146798277467802655432601, 9.965932715487020178134239291621

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