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Properties

Label 2-1375-1375.769-c0-0-0
Degree $2$
Conductor $1375$
Sign $0.675 - 0.737i$
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  + (0.503 + 0.536i)3-s + (0.425 + 0.904i)4-s + (0.809 − 0.587i)5-s + (0.0287 − 0.457i)9-s + (−0.535 + 0.844i)11-s + (−0.271 + 0.684i)12-s + (0.723 + 0.137i)15-s + (−0.637 + 0.770i)16-s + (0.876 + 0.481i)20-s + (−0.340 + 1.32i)23-s + (0.309 − 0.951i)25-s + (0.827 − 0.684i)27-s + (0.824 − 1.75i)31-s + (−0.723 + 0.137i)33-s + (0.425 − 0.168i)36-s + (−1.46 − 1.21i)37-s + ⋯
L(s)  = 1  + (0.503 + 0.536i)3-s + (0.425 + 0.904i)4-s + (0.809 − 0.587i)5-s + (0.0287 − 0.457i)9-s + (−0.535 + 0.844i)11-s + (−0.271 + 0.684i)12-s + (0.723 + 0.137i)15-s + (−0.637 + 0.770i)16-s + (0.876 + 0.481i)20-s + (−0.340 + 1.32i)23-s + (0.309 − 0.951i)25-s + (0.827 − 0.684i)27-s + (0.824 − 1.75i)31-s + (−0.723 + 0.137i)33-s + (0.425 − 0.168i)36-s + (−1.46 − 1.21i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.675 - 0.737i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.675 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.540379992\)
\(L(\frac12)\) \(\approx\) \(1.540379992\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.535 - 0.844i)T \)
good2 \( 1 + (-0.425 - 0.904i)T^{2} \)
3 \( 1 + (-0.503 - 0.536i)T + (-0.0627 + 0.998i)T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.992 - 0.125i)T^{2} \)
17 \( 1 + (-0.637 - 0.770i)T^{2} \)
19 \( 1 + (-0.0627 - 0.998i)T^{2} \)
23 \( 1 + (0.340 - 1.32i)T + (-0.876 - 0.481i)T^{2} \)
29 \( 1 + (-0.535 - 0.844i)T^{2} \)
31 \( 1 + (-0.824 + 1.75i)T + (-0.637 - 0.770i)T^{2} \)
37 \( 1 + (1.46 + 1.21i)T + (0.187 + 0.982i)T^{2} \)
41 \( 1 + (-0.876 + 0.481i)T^{2} \)
43 \( 1 + (-0.809 - 0.587i)T^{2} \)
47 \( 1 + (-0.147 - 1.16i)T + (-0.968 + 0.248i)T^{2} \)
53 \( 1 + (1.65 + 0.316i)T + (0.929 + 0.368i)T^{2} \)
59 \( 1 + (-1.18 - 0.469i)T + (0.728 + 0.684i)T^{2} \)
61 \( 1 + (-0.876 - 0.481i)T^{2} \)
67 \( 1 + (0.946 + 1.72i)T + (-0.535 + 0.844i)T^{2} \)
71 \( 1 + (-1.84 + 0.233i)T + (0.968 - 0.248i)T^{2} \)
73 \( 1 + (0.728 - 0.684i)T^{2} \)
79 \( 1 + (-0.0627 + 0.998i)T^{2} \)
83 \( 1 + (0.0627 + 0.998i)T^{2} \)
89 \( 1 + (1.84 - 0.730i)T + (0.728 - 0.684i)T^{2} \)
97 \( 1 + (-0.742 + 1.35i)T + (-0.535 - 0.844i)T^{2} \)
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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

−9.518773939998228407789932441495, −9.362976600776658631264363172547, −8.254403159104601559233304887445, −7.65994561071790699978351065168, −6.65031181352188556362135366553, −5.75062940569210472819215740919, −4.64652924113401384196822444774, −3.85253673452311152725403363374, −2.81746142935230781327825623306, −1.86313076305075860826946815005, 1.45775368225439779630557728360, 2.41287710661874078143059109565, 3.13907776281258291100371356950, 4.93307729114571056838983258780, 5.54868999890776840126866781081, 6.65881040046516439467380796464, 6.87563486317750973211540405373, 8.194932324010614586071675079704, 8.714147668912521163644572551718, 9.934771506949825503784407922112

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