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Properties

Label 2-1035-15.8-c1-0-34
Degree $2$
Conductor $1035$
Sign $0.0894 + 0.995i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0894 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0894 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $0.0894 + 0.995i$
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: \((2,\ 1035,\ (\ :1/2),\ 0.0894 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.024711160\)
\(L(\frac12)\) \(\approx\) \(2.024711160\)
\(L(\frac{3}{2})\) 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

−9.916752123404820881130837510133, −8.867777461185405549096851664600, −8.358851595589408131423654100474, −6.99150434463077123622745680517, −6.14937258895287706668409848585, −5.49447812807906780535580658824, −4.74392636220092716731386912694, −3.44135950785990833162250616744, −1.88447589673777773297011947978, −0.962027322321833479712480788627, 1.79154270430315016530768838190, 2.78478330825091964770261502526, 3.74665869056195712623188758868, 5.00285074161315596001931050093, 5.75316691945402604283876355438, 6.99094489920882209860542249992, 7.50726830444334554894434694705, 8.459996424728184165282178399066, 9.363241275093281239864348830289, 10.05665289301285628580699331706

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