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Properties

Label 2-1035-15.2-c1-0-31
Degree $2$
Conductor $1035$
Sign $0.862 - 0.506i$
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  + (−1.27 + 1.27i)2-s − 1.24i·4-s + (2.19 + 0.428i)5-s + (0.936 + 0.936i)7-s + (−0.958 − 0.958i)8-s + (−3.34 + 2.25i)10-s − 0.533i·11-s + (4.37 − 4.37i)13-s − 2.38·14-s + 4.93·16-s + (0.921 − 0.921i)17-s − 7.99i·19-s + (0.534 − 2.73i)20-s + (0.679 + 0.679i)22-s + (0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (−0.901 + 0.901i)2-s − 0.623i·4-s + (0.981 + 0.191i)5-s + (0.354 + 0.354i)7-s + (−0.339 − 0.339i)8-s + (−1.05 + 0.711i)10-s − 0.160i·11-s + (1.21 − 1.21i)13-s − 0.637·14-s + 1.23·16-s + (0.223 − 0.223i)17-s − 1.83i·19-s + (0.119 − 0.612i)20-s + (0.144 + 0.144i)22-s + (0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $0.862 - 0.506i$
Analytic conductor: \(8.26451\)
Root analytic conductor: \(2.87480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1035,\ (\ :1/2),\ 0.862 - 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.230942934\)
\(L(\frac12)\) \(\approx\) \(1.230942934\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.19 - 0.428i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.27 - 1.27i)T - 2iT^{2} \)
7 \( 1 + (-0.936 - 0.936i)T + 7iT^{2} \)
11 \( 1 + 0.533iT - 11T^{2} \)
13 \( 1 + (-4.37 + 4.37i)T - 13iT^{2} \)
17 \( 1 + (-0.921 + 0.921i)T - 17iT^{2} \)
19 \( 1 + 7.99iT - 19T^{2} \)
29 \( 1 + 9.15T + 29T^{2} \)
31 \( 1 - 2.22T + 31T^{2} \)
37 \( 1 + (2.15 + 2.15i)T + 37iT^{2} \)
41 \( 1 + 11.2iT - 41T^{2} \)
43 \( 1 + (-5.40 + 5.40i)T - 43iT^{2} \)
47 \( 1 + (-1.53 + 1.53i)T - 47iT^{2} \)
53 \( 1 + (-2.98 - 2.98i)T + 53iT^{2} \)
59 \( 1 - 6.39T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 + (-9.65 - 9.65i)T + 67iT^{2} \)
71 \( 1 - 1.85iT - 71T^{2} \)
73 \( 1 + (9.68 - 9.68i)T - 73iT^{2} \)
79 \( 1 - 9.38iT - 79T^{2} \)
83 \( 1 + (-8.08 - 8.08i)T + 83iT^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 + (-1.22 - 1.22i)T + 97iT^{2} \)
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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

−9.737123924103716132086640663461, −8.888161788542107183448860317199, −8.596870221278642026479517444704, −7.42851161790277001358695864374, −6.84139014023610837659217582161, −5.69834944187816479353165165334, −5.44528632049726142271948865835, −3.64278991076580745693713011552, −2.46735255605092837845052272531, −0.850900883138334752211473905292, 1.39596709432730384418454545630, 1.81940704811195505275446238190, 3.29049688837537469589022811385, 4.43618620346846538267546289077, 5.77915583853925707409563684440, 6.29603365969416887573201842175, 7.70666454198643902993437430145, 8.492049591480155416376651728108, 9.291760539148091601468484174690, 9.800894580019244157590834519643

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