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Properties

Label 2-1035-115.34-c0-0-1
Degree $2$
Conductor $1035$
Sign $-0.874 - 0.484i$
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  + (−1.07 − 1.66i)2-s + (−1.21 + 2.65i)4-s + (−0.959 + 0.281i)5-s + (3.76 − 0.540i)8-s + (1.49 + 1.29i)10-s + (−3.01 − 3.47i)16-s + (−0.345 − 0.755i)17-s + (−1.37 − 0.627i)19-s + (0.415 − 2.88i)20-s + (−0.841 − 0.540i)23-s + (0.841 − 0.540i)25-s + (−0.186 − 1.29i)31-s + (−1.49 + 5.09i)32-s + (−0.889 + 1.38i)34-s + (0.425 + 2.96i)38-s + ⋯
L(s)  = 1  + (−1.07 − 1.66i)2-s + (−1.21 + 2.65i)4-s + (−0.959 + 0.281i)5-s + (3.76 − 0.540i)8-s + (1.49 + 1.29i)10-s + (−3.01 − 3.47i)16-s + (−0.345 − 0.755i)17-s + (−1.37 − 0.627i)19-s + (0.415 − 2.88i)20-s + (−0.841 − 0.540i)23-s + (0.841 − 0.540i)25-s + (−0.186 − 1.29i)31-s + (−1.49 + 5.09i)32-s + (−0.889 + 1.38i)34-s + (0.425 + 2.96i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $-0.874 - 0.484i$
Analytic conductor: \(0.516532\)
Root analytic conductor: \(0.718701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1035,\ (\ :0),\ -0.874 - 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2184584080\)
\(L(\frac12)\) \(\approx\) \(0.2184584080\)
\(L(1)\) 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 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
good2 \( 1 + (1.07 + 1.66i)T + (-0.415 + 0.909i)T^{2} \)
7 \( 1 + (-0.142 + 0.989i)T^{2} \)
11 \( 1 + (-0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.142 + 0.989i)T^{2} \)
17 \( 1 + (0.345 + 0.755i)T + (-0.654 + 0.755i)T^{2} \)
19 \( 1 + (1.37 + 0.627i)T + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (-0.654 + 0.755i)T^{2} \)
31 \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \)
37 \( 1 + (0.841 + 0.540i)T^{2} \)
41 \( 1 + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (-0.959 - 0.281i)T^{2} \)
47 \( 1 - 1.08iT - T^{2} \)
53 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (1.07 - 0.153i)T + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (0.425 + 0.368i)T + (0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.841 - 0.540i)T^{2} \)
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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

−9.774050379559264785778925874880, −8.959927304988823753249698351443, −8.253161014959665000797905502987, −7.61113835658180741986047820332, −6.65688751056734141560086723345, −4.63241848854074474056744820023, −4.03895152004890064045443059973, −2.98076015810112659231509884997, −2.08939265985803490993156570244, −0.29610066118754624069314019043, 1.57706190386261152032571997430, 3.97386807488616327472987822238, 4.77357240098153922524798364969, 5.84583493832856951881473504578, 6.56521378366491621612206425272, 7.44202184236778392749718721948, 8.140057810051660764589355190455, 8.639073411155034599709506349032, 9.405385474724249998102063936276, 10.47752249866333772565391604694

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