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Properties

Label 2-1395-155.29-c0-0-1
Degree $2$
Conductor $1395$
Sign $0.795 + 0.606i$
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  + (0.363 − 0.5i)2-s + (0.190 + 0.587i)4-s − i·5-s + (0.951 + 0.309i)8-s + (−0.5 − 0.363i)10-s + (1.30 + 0.951i)19-s + (0.587 − 0.190i)20-s + (0.587 − 1.80i)23-s − 25-s + (−0.809 − 0.587i)31-s + i·32-s + (0.951 − 0.309i)38-s + (0.309 − 0.951i)40-s + (−0.690 − 0.951i)46-s + (0.363 + 0.5i)47-s + ⋯
L(s)  = 1  + (0.363 − 0.5i)2-s + (0.190 + 0.587i)4-s − i·5-s + (0.951 + 0.309i)8-s + (−0.5 − 0.363i)10-s + (1.30 + 0.951i)19-s + (0.587 − 0.190i)20-s + (0.587 − 1.80i)23-s − 25-s + (−0.809 − 0.587i)31-s + i·32-s + (0.951 − 0.309i)38-s + (0.309 − 0.951i)40-s + (−0.690 − 0.951i)46-s + (0.363 + 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1395\)    =    \(3^{2} \cdot 5 \cdot 31\)
Sign: $0.795 + 0.606i$
Analytic conductor: \(0.696195\)
Root analytic conductor: \(0.834383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1395} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1395,\ (\ :0),\ 0.795 + 0.606i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.463034083\)
\(L(\frac12)\) \(\approx\) \(1.463034083\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + 1.17iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.646747024033676392443758834221, −8.871558305370632235056718977394, −8.034005020074419710876804462623, −7.48959411315008385316825199420, −6.32349188929028102429988387223, −5.26273645986140265933938842152, −4.50065485956841016711323034111, −3.66041670334128565156307650565, −2.60349519890211859919876146798, −1.37996139760163965698057665027, 1.54494309668194246395939242961, 2.88540190191401081781024018776, 3.83229086827718138356579869011, 5.16759512364845796550080002980, 5.60046946649591762409196702439, 6.76858494443813440152724753447, 7.11258643077284693217677869883, 7.902589219178824442689613725168, 9.287199837529105536289295676390, 9.791594188564955742501809665646

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