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Properties

Label 2-1375-1375.1341-c0-0-0
Degree $2$
Conductor $1375$
Sign $0.988 + 0.150i$
Analytic cond. $0.686214$
Root an. cond. $0.828380$
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  + (0.939 − 1.47i)3-s + (0.0627 + 0.998i)4-s + (0.309 + 0.951i)5-s + (−0.882 − 1.87i)9-s + (0.728 + 0.684i)11-s + (1.53 + 0.844i)12-s + (1.69 + 0.435i)15-s + (−0.992 + 0.125i)16-s + (−0.929 + 0.368i)20-s + (−0.200 − 1.05i)23-s + (−0.809 + 0.587i)25-s + (−1.86 − 0.235i)27-s + (−0.0235 + 0.374i)31-s + (1.69 − 0.435i)33-s + (1.81 − 0.998i)36-s + (1.60 − 0.202i)37-s + ⋯
L(s)  = 1  + (0.939 − 1.47i)3-s + (0.0627 + 0.998i)4-s + (0.309 + 0.951i)5-s + (−0.882 − 1.87i)9-s + (0.728 + 0.684i)11-s + (1.53 + 0.844i)12-s + (1.69 + 0.435i)15-s + (−0.992 + 0.125i)16-s + (−0.929 + 0.368i)20-s + (−0.200 − 1.05i)23-s + (−0.809 + 0.587i)25-s + (−1.86 − 0.235i)27-s + (−0.0235 + 0.374i)31-s + (1.69 − 0.435i)33-s + (1.81 − 0.998i)36-s + (1.60 − 0.202i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.558299706\)
\(L(\frac12)\) \(\approx\) \(1.558299706\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.728 - 0.684i)T \)
good2 \( 1 + (-0.0627 - 0.998i)T^{2} \)
3 \( 1 + (-0.939 + 1.47i)T + (-0.425 - 0.904i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.637 - 0.770i)T^{2} \)
17 \( 1 + (0.992 + 0.125i)T^{2} \)
19 \( 1 + (0.425 - 0.904i)T^{2} \)
23 \( 1 + (0.200 + 1.05i)T + (-0.929 + 0.368i)T^{2} \)
29 \( 1 + (-0.728 + 0.684i)T^{2} \)
31 \( 1 + (0.0235 - 0.374i)T + (-0.992 - 0.125i)T^{2} \)
37 \( 1 + (-1.60 + 0.202i)T + (0.968 - 0.248i)T^{2} \)
41 \( 1 + (0.929 + 0.368i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.393 + 0.476i)T + (-0.187 + 0.982i)T^{2} \)
53 \( 1 + (-1.41 - 0.362i)T + (0.876 + 0.481i)T^{2} \)
59 \( 1 + (1.73 + 0.955i)T + (0.535 + 0.844i)T^{2} \)
61 \( 1 + (0.929 - 0.368i)T^{2} \)
67 \( 1 + (1.80 + 0.713i)T + (0.728 + 0.684i)T^{2} \)
71 \( 1 + (1.11 + 1.35i)T + (-0.187 + 0.982i)T^{2} \)
73 \( 1 + (-0.535 + 0.844i)T^{2} \)
79 \( 1 + (0.425 + 0.904i)T^{2} \)
83 \( 1 + (0.425 - 0.904i)T^{2} \)
89 \( 1 + (1.11 - 0.614i)T + (0.535 - 0.844i)T^{2} \)
97 \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \)
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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

−9.419870826446896196840348025407, −8.834509207778856114769881290689, −7.83611593147339597739303947198, −7.49335427654205429060116673033, −6.62884571640879088318793419758, −6.25289122205494290763865640886, −4.36964490454421055153038242977, −3.29248260790176669546478211967, −2.59868115103302100603850604061, −1.74953871968524199025171124667, 1.47040419388496241360827868700, 2.79693321554972873010274636500, 3.99599540109971546510428998817, 4.60270409678896644432252108082, 5.51308258032386698520131100077, 6.12970683858177792480547020292, 7.64809650039598722612800599275, 8.680657150643804557474045291395, 9.064485082481912234705636213039, 9.768331342282767955464286812890

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