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Properties

Label 2-1375-275.131-c0-0-2
Degree $2$
Conductor $1375$
Sign $0.436 + 0.899i$
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  + (1.53 − 1.11i)3-s + (−0.809 + 0.587i)4-s + (0.809 − 2.48i)9-s + (−0.309 − 0.951i)11-s + (−0.587 + 1.80i)12-s + (0.309 − 0.951i)16-s + (0.363 + 1.11i)23-s + (−0.951 − 2.92i)27-s + (0.5 + 0.363i)31-s + (−1.53 − 1.11i)33-s + (0.809 + 2.48i)36-s + (0.809 + 0.587i)44-s + (−0.587 − 1.80i)48-s + 49-s + (−1.53 + 1.11i)53-s + ⋯
L(s)  = 1  + (1.53 − 1.11i)3-s + (−0.809 + 0.587i)4-s + (0.809 − 2.48i)9-s + (−0.309 − 0.951i)11-s + (−0.587 + 1.80i)12-s + (0.309 − 0.951i)16-s + (0.363 + 1.11i)23-s + (−0.951 − 2.92i)27-s + (0.5 + 0.363i)31-s + (−1.53 − 1.11i)33-s + (0.809 + 2.48i)36-s + (0.809 + 0.587i)44-s + (−0.587 − 1.80i)48-s + 49-s + (−1.53 + 1.11i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.436 + 0.899i$
Analytic conductor: \(0.686214\)
Root analytic conductor: \(0.828380\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1375,\ (\ :0),\ 0.436 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.479367874\)
\(L(\frac12)\) \(\approx\) \(1.479367874\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.173495967444163016952059808512, −8.802200517775485769897755004978, −7.994111113616083940774081628971, −7.57760657201761314509896336462, −6.65904129994098808913630964971, −5.53448785215072197769526863363, −4.16528458426794773293799259974, −3.30464684300679108315079506559, −2.68674765846220288117169326037, −1.21071307759098261354028857211, 1.93037705094594083938231811258, 2.96590718514139981671025133650, 4.07916707876139500936459124123, 4.61401293765354839415951131233, 5.35783220200024091871986110636, 6.78472483612592532494503681763, 7.956766696489681844923259084255, 8.419314229170431553554973004976, 9.324135297849864696841582983277, 9.684584710183209795674979111697

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