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Properties

Label 2-1035-15.2-c1-0-1
Degree $2$
Conductor $1035$
Sign $-0.968 + 0.247i$
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  + (−0.0986 + 0.0986i)2-s + 1.98i·4-s + (−1.99 − 1.01i)5-s + (0.908 + 0.908i)7-s + (−0.392 − 0.392i)8-s + (0.296 − 0.0957i)10-s − 3.38i·11-s + (0.942 − 0.942i)13-s − 0.179·14-s − 3.88·16-s + (−5.15 + 5.15i)17-s + 4.57i·19-s + (2.01 − 3.94i)20-s + (0.333 + 0.333i)22-s + (0.707 + 0.707i)23-s + ⋯
L(s)  = 1  + (−0.0697 + 0.0697i)2-s + 0.990i·4-s + (−0.890 − 0.455i)5-s + (0.343 + 0.343i)7-s + (−0.138 − 0.138i)8-s + (0.0938 − 0.0302i)10-s − 1.02i·11-s + (0.261 − 0.261i)13-s − 0.0478·14-s − 0.970·16-s + (−1.25 + 1.25i)17-s + 1.04i·19-s + (0.451 − 0.881i)20-s + (0.0711 + 0.0711i)22-s + (0.147 + 0.147i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $-0.968 + 0.247i$
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: \((2,\ 1035,\ (\ :1/2),\ -0.968 + 0.247i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1899572405\)
\(L(\frac12)\) \(\approx\) \(0.1899572405\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.99 + 1.01i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (0.0986 - 0.0986i)T - 2iT^{2} \)
7 \( 1 + (-0.908 - 0.908i)T + 7iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (-0.942 + 0.942i)T - 13iT^{2} \)
17 \( 1 + (5.15 - 5.15i)T - 17iT^{2} \)
19 \( 1 - 4.57iT - 19T^{2} \)
29 \( 1 + 4.70T + 29T^{2} \)
31 \( 1 + 6.77T + 31T^{2} \)
37 \( 1 + (8.21 + 8.21i)T + 37iT^{2} \)
41 \( 1 + 0.370iT - 41T^{2} \)
43 \( 1 + (-0.155 + 0.155i)T - 43iT^{2} \)
47 \( 1 + (-6.67 + 6.67i)T - 47iT^{2} \)
53 \( 1 + (7.54 + 7.54i)T + 53iT^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 + 9.21T + 61T^{2} \)
67 \( 1 + (-0.394 - 0.394i)T + 67iT^{2} \)
71 \( 1 - 6.01iT - 71T^{2} \)
73 \( 1 + (0.849 - 0.849i)T - 73iT^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 + (1.16 + 1.16i)T + 83iT^{2} \)
89 \( 1 - 8.53T + 89T^{2} \)
97 \( 1 + (-9.92 - 9.92i)T + 97iT^{2} \)
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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

−10.70057098149768975891852072046, −9.047018391483443642374817871541, −8.685795248641309732018307775521, −8.008821539095874423745680922396, −7.29466724691248610984963453759, −6.15844036615241917678274100581, −5.16389349995365953846721818378, −3.84104996175354824616107116829, −3.58415509360818072212987996878, −1.94509559802039975451691457074, 0.084451153398113415040093539834, 1.73679112214212240391959033781, 2.96881585278643219518219927737, 4.53723551063104751057119731202, 4.74887662080825904319494012386, 6.21994918670711585880367393109, 7.09107660700729729587404245345, 7.51900372475084197821087383536, 8.974802369692398626990641165772, 9.336988040877562480687377116266

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