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Properties

Label 2-1035-15.8-c1-0-16
Degree $2$
Conductor $1035$
Sign $-0.500 - 0.865i$
Analytic cond. $8.26451$
Root an. cond. $2.87480$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.56 + 1.56i)2-s + 2.91i·4-s + (−2.12 − 0.708i)5-s + (1.66 − 1.66i)7-s + (−1.42 + 1.42i)8-s + (−2.21 − 4.43i)10-s + 6.00i·11-s + (0.950 + 0.950i)13-s + 5.21·14-s + 1.35·16-s + (3.39 + 3.39i)17-s + 4.77i·19-s + (2.06 − 6.17i)20-s + (−9.41 + 9.41i)22-s + (0.707 − 0.707i)23-s + ⋯
L(s)  = 1  + (1.10 + 1.10i)2-s + 1.45i·4-s + (−0.948 − 0.317i)5-s + (0.629 − 0.629i)7-s + (−0.504 + 0.504i)8-s + (−0.699 − 1.40i)10-s + 1.81i·11-s + (0.263 + 0.263i)13-s + 1.39·14-s + 0.337·16-s + (0.822 + 0.822i)17-s + 1.09i·19-s + (0.461 − 1.38i)20-s + (−2.00 + 2.00i)22-s + (0.147 − 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $-0.500 - 0.865i$
Analytic conductor: \(8.26451\)
Root analytic conductor: \(2.87480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1035,\ (\ :1/2),\ -0.500 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.623350017\)
\(L(\frac12)\) \(\approx\) \(2.623350017\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 + 0.708i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-1.56 - 1.56i)T + 2iT^{2} \)
7 \( 1 + (-1.66 + 1.66i)T - 7iT^{2} \)
11 \( 1 - 6.00iT - 11T^{2} \)
13 \( 1 + (-0.950 - 0.950i)T + 13iT^{2} \)
17 \( 1 + (-3.39 - 3.39i)T + 17iT^{2} \)
19 \( 1 - 4.77iT - 19T^{2} \)
29 \( 1 + 8.39T + 29T^{2} \)
31 \( 1 + 6.12T + 31T^{2} \)
37 \( 1 + (-8.08 + 8.08i)T - 37iT^{2} \)
41 \( 1 - 3.39iT - 41T^{2} \)
43 \( 1 + (3.81 + 3.81i)T + 43iT^{2} \)
47 \( 1 + (-4.67 - 4.67i)T + 47iT^{2} \)
53 \( 1 + (-1.97 + 1.97i)T - 53iT^{2} \)
59 \( 1 - 0.951T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 + (-4.40 + 4.40i)T - 67iT^{2} \)
71 \( 1 + 16.3iT - 71T^{2} \)
73 \( 1 + (8.01 + 8.01i)T + 73iT^{2} \)
79 \( 1 + 6.34iT - 79T^{2} \)
83 \( 1 + (3.16 - 3.16i)T - 83iT^{2} \)
89 \( 1 + 4.16T + 89T^{2} \)
97 \( 1 + (3.80 - 3.80i)T - 97iT^{2} \)
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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

−10.26263298022505719192049598875, −9.207575803241447314975228484258, −7.83571907827761474928121008734, −7.71051603456209148155157573118, −6.95932360731274062762305863040, −5.81260992987609726499172681392, −4.96554878378445346058107586503, −4.09035012732589176951428185100, −3.76210809903355739221162206520, −1.65823003156160050694212122640, 0.914279912824269191999634907890, 2.57874997363657562959100693253, 3.28744472509482192957672744243, 4.06551113992823686952071156919, 5.26841515350367186970360996538, 5.68559961654372676062239922803, 7.09800997076261019725542679173, 8.127023215903795805878873369634, 8.778190889562796299387233055467, 9.977165646015800356259863086522

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