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Properties

Label 2-1035-115.84-c0-0-1
Degree $2$
Conductor $1035$
Sign $0.991 - 0.128i$
Analytic cond. $0.516532$
Root an. cond. $0.718701$
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  + (1.80 + 0.258i)2-s + (2.21 + 0.650i)4-s + (−0.654 − 0.755i)5-s + (2.16 + 0.989i)8-s + (−0.983 − 1.53i)10-s + (1.70 + 1.09i)16-s + (−1.84 + 0.540i)17-s + (0.304 − 1.03i)19-s + (−0.959 − 2.10i)20-s + (0.142 + 0.989i)23-s + (−0.142 + 0.989i)25-s + (−0.698 + 1.53i)31-s + (0.983 + 0.852i)32-s + (−3.45 + 0.496i)34-s + (0.817 − 1.78i)38-s + ⋯
L(s)  = 1  + (1.80 + 0.258i)2-s + (2.21 + 0.650i)4-s + (−0.654 − 0.755i)5-s + (2.16 + 0.989i)8-s + (−0.983 − 1.53i)10-s + (1.70 + 1.09i)16-s + (−1.84 + 0.540i)17-s + (0.304 − 1.03i)19-s + (−0.959 − 2.10i)20-s + (0.142 + 0.989i)23-s + (−0.142 + 0.989i)25-s + (−0.698 + 1.53i)31-s + (0.983 + 0.852i)32-s + (−3.45 + 0.496i)34-s + (0.817 − 1.78i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $0.991 - 0.128i$
Analytic conductor: \(0.516532\)
Root analytic conductor: \(0.718701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1035,\ (\ :0),\ 0.991 - 0.128i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.491310068\)
\(L(\frac12)\) \(\approx\) \(2.491310068\)
\(L(1)\) 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 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (-0.142 - 0.989i)T \)
good2 \( 1 + (-1.80 - 0.258i)T + (0.959 + 0.281i)T^{2} \)
7 \( 1 + (0.415 + 0.909i)T^{2} \)
11 \( 1 + (0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (1.84 - 0.540i)T + (0.841 - 0.540i)T^{2} \)
19 \( 1 + (-0.304 + 1.03i)T + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (0.841 - 0.540i)T^{2} \)
31 \( 1 + (0.698 - 1.53i)T + (-0.654 - 0.755i)T^{2} \)
37 \( 1 + (-0.142 - 0.989i)T^{2} \)
41 \( 1 + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + 1.97iT - T^{2} \)
53 \( 1 + (0.239 + 0.153i)T + (0.415 + 0.909i)T^{2} \)
59 \( 1 + (0.415 - 0.909i)T^{2} \)
61 \( 1 + (-1.80 - 0.822i)T + (0.654 + 0.755i)T^{2} \)
67 \( 1 + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (-0.841 - 0.540i)T^{2} \)
79 \( 1 + (0.817 + 1.27i)T + (-0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
89 \( 1 + (0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.142 + 0.989i)T^{2} \)
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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

−10.55224534005618095408529924757, −9.095259419937350329248844691130, −8.381232453526720123577737882518, −7.16282845570123941714011314724, −6.76984685727074212721051762151, −5.50880072714793806390971910791, −4.91388920425011754209917282449, −4.10531202873362421567130734364, −3.30544461803083701405812679942, −1.97705588049490233501847823368, 2.18157508823499889326832636675, 2.99996059074540101041788821585, 4.06796445628218171972533343776, 4.55927289572106884568221646098, 5.79173827247193279844491984599, 6.51943916388485339046318162826, 7.21117313246775457280480793604, 8.166365537236139904474465600573, 9.519929491081151138963140881684, 10.65174671496911118122501229763

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