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Properties

Label 2-990-15.8-c1-0-16
Degree $2$
Conductor $990$
Sign $0.995 + 0.0930i$
Analytic cond. $7.90518$
Root an. cond. $2.81161$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.80 − 1.31i)5-s + (3.12 − 3.12i)7-s + (−0.707 + 0.707i)8-s + (2.20 + 0.345i)10-s + i·11-s + (−0.0741 − 0.0741i)13-s + 4.41·14-s − 1.00·16-s + (−2.26 − 2.26i)17-s − 0.608i·19-s + (1.31 + 1.80i)20-s + (−0.707 + 0.707i)22-s + (2.33 − 2.33i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.807 − 0.589i)5-s + (1.18 − 1.18i)7-s + (−0.250 + 0.250i)8-s + (0.698 + 0.109i)10-s + 0.301i·11-s + (−0.0205 − 0.0205i)13-s + 1.18·14-s − 0.250·16-s + (−0.548 − 0.548i)17-s − 0.139i·19-s + (0.294 + 0.403i)20-s + (−0.150 + 0.150i)22-s + (0.486 − 0.486i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(990\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $0.995 + 0.0930i$
Analytic conductor: \(7.90518\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{990} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 990,\ (\ :1/2),\ 0.995 + 0.0930i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63006 - 0.122602i\)
\(L(\frac12)\) \(\approx\) \(2.63006 - 0.122602i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (-1.80 + 1.31i)T \)
11 \( 1 - iT \)
good7 \( 1 + (-3.12 + 3.12i)T - 7iT^{2} \)
13 \( 1 + (0.0741 + 0.0741i)T + 13iT^{2} \)
17 \( 1 + (2.26 + 2.26i)T + 17iT^{2} \)
19 \( 1 + 0.608iT - 19T^{2} \)
23 \( 1 + (-2.33 + 2.33i)T - 23iT^{2} \)
29 \( 1 + 1.78T + 29T^{2} \)
31 \( 1 + 4.66T + 31T^{2} \)
37 \( 1 + (-6.04 + 6.04i)T - 37iT^{2} \)
41 \( 1 - 11.6iT - 41T^{2} \)
43 \( 1 + (-2.89 - 2.89i)T + 43iT^{2} \)
47 \( 1 + (-1.82 - 1.82i)T + 47iT^{2} \)
53 \( 1 + (7.79 - 7.79i)T - 53iT^{2} \)
59 \( 1 + 3.13T + 59T^{2} \)
61 \( 1 - 14.5T + 61T^{2} \)
67 \( 1 + (-0.791 + 0.791i)T - 67iT^{2} \)
71 \( 1 - 16.1iT - 71T^{2} \)
73 \( 1 + (-1.67 - 1.67i)T + 73iT^{2} \)
79 \( 1 - 1.69iT - 79T^{2} \)
83 \( 1 + (6.00 - 6.00i)T - 83iT^{2} \)
89 \( 1 - 0.539T + 89T^{2} \)
97 \( 1 + (8.80 - 8.80i)T - 97iT^{2} \)
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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

−9.886407514423304051336558894619, −9.085400518562321359412310646722, −8.144543518386634583872221083679, −7.41811719718981734301444552872, −6.59331078875660218715224914053, −5.50824187983746317840062921883, −4.70443430896099680546935556154, −4.14538404440720094833566295449, −2.49158370418655934344117860423, −1.16916423266595455464405344341, 1.71278391997396938467862305048, 2.39050388669600033281243765402, 3.57174454805646907430560033627, 4.90439273095740082508132975274, 5.57714414685324813321606684832, 6.28577139196835077708724136051, 7.43206030186136388713829397897, 8.574318458559657960445394869163, 9.185192135047452964505999624208, 10.14157370557413317251390226257

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