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Properties

Label 2-1395-155.89-c0-0-1
Degree $2$
Conductor $1395$
Sign $0.358 + 0.933i$
Analytic cond. $0.696195$
Root an. cond. $0.834383$
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 − 0.5i)2-s + (1.30 − 0.951i)4-s − i·5-s + (0.587 − 0.809i)8-s + (−0.5 − 1.53i)10-s + (0.190 + 0.587i)19-s + (−0.951 − 1.30i)20-s + (−0.951 − 0.690i)23-s − 25-s + (0.309 + 0.951i)31-s + 0.999i·32-s + (0.587 + 0.809i)38-s + (−0.809 − 0.587i)40-s + (−1.80 − 0.587i)46-s + (1.53 + 0.5i)47-s + ⋯
L(s)  = 1  + (1.53 − 0.5i)2-s + (1.30 − 0.951i)4-s − i·5-s + (0.587 − 0.809i)8-s + (−0.5 − 1.53i)10-s + (0.190 + 0.587i)19-s + (−0.951 − 1.30i)20-s + (−0.951 − 0.690i)23-s − 25-s + (0.309 + 0.951i)31-s + 0.999i·32-s + (0.587 + 0.809i)38-s + (−0.809 − 0.587i)40-s + (−1.80 − 0.587i)46-s + (1.53 + 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1395\)    =    \(3^{2} \cdot 5 \cdot 31\)
Sign: $0.358 + 0.933i$
Analytic conductor: \(0.696195\)
Root analytic conductor: \(0.834383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1395} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1395,\ (\ :0),\ 0.358 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.409741043\)
\(L(\frac12)\) \(\approx\) \(2.409741043\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + iT \)
31 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (-0.809 + 0.587i)T^{2} \)
47 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 - 1.90iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−9.818684069881414259302465130964, −8.753983740071987180946283332500, −8.064501697644295898606411874298, −6.85350523312513707046930975077, −5.89534122371692675179987885146, −5.30263693069996290297020374787, −4.42067735165939050367518747074, −3.81178444949161668008886221696, −2.63741126863870729218807037891, −1.51567127038477989358590995554, 2.25850831761372306296630769405, 3.18166569176664702405429777885, 3.97785921922411868494987377839, 4.86292823119690529805409648204, 5.94159451904908591287639136531, 6.30308798844611250287930052410, 7.35963804934999361720053819444, 7.74163580380458491175999563514, 9.161550232365969259682412688424, 10.02334481070636130257011449793

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