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Properties

Label 2-1352-104.43-c0-0-7
Degree $2$
Conductor $1352$
Sign $-0.711 + 0.702i$
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 + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.499 − 0.866i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (0.5 − 0.866i)10-s − 0.999·12-s − 0.999·14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)20-s − 0.999·21-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (−0.499 − 0.866i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (0.5 − 0.866i)10-s − 0.999·12-s − 0.999·14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.499 − 0.866i)20-s − 0.999·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ -0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.693740781\)
\(L(\frac12)\) \(\approx\) \(1.693740781\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.656361258447832361351970541212, −8.901489103286226464715055778822, −7.85641125598285750583854574681, −6.95082644074089100118944442157, −6.13245980298341720261497565501, −5.32785995138195198531269442945, −4.11916395240540342828665887712, −3.19189314768370573786091537147, −2.09984144407098855652649634303, −1.33863083999387774122270682158, 2.36510886821755244404225231322, 3.29754264747014878341025719717, 4.16073480727035636401139162302, 5.40223010768282246688390057095, 5.66789066664759067689681095834, 6.73673343551487987721213397314, 7.55898338060417101902618265648, 8.821906459645765264092299128194, 9.176236979548475948595703556973, 9.694424961088856943637123830944

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