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Properties

Label 2-1190-85.84-c1-0-36
Degree $2$
Conductor $1190$
Sign $0.654 - 0.755i$
Analytic cond. $9.50219$
Root an. cond. $3.08256$
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  + i·2-s + 2.54·3-s − 4-s + (0.885 − 2.05i)5-s + 2.54i·6-s + 7-s − i·8-s + 3.46·9-s + (2.05 + 0.885i)10-s + 6.12i·11-s − 2.54·12-s + 4.24i·13-s + i·14-s + (2.25 − 5.22i)15-s + 16-s + (1.24 − 3.93i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.46·3-s − 0.5·4-s + (0.396 − 0.918i)5-s + 1.03i·6-s + 0.377·7-s − 0.353i·8-s + 1.15·9-s + (0.649 + 0.280i)10-s + 1.84i·11-s − 0.734·12-s + 1.17i·13-s + 0.267i·14-s + (0.581 − 1.34i)15-s + 0.250·16-s + (0.301 − 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1190\)    =    \(2 \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(9.50219\)
Root analytic conductor: \(3.08256\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1190} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1190,\ (\ :1/2),\ 0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.908871574\)
\(L(\frac12)\) \(\approx\) \(2.908871574\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-0.885 + 2.05i)T \)
7 \( 1 - T \)
17 \( 1 + (-1.24 + 3.93i)T \)
good3 \( 1 - 2.54T + 3T^{2} \)
11 \( 1 - 6.12iT - 11T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
19 \( 1 - 5.07T + 19T^{2} \)
23 \( 1 + 1.01T + 23T^{2} \)
29 \( 1 + 1.11iT - 29T^{2} \)
31 \( 1 - 1.09iT - 31T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 + 6.80iT - 41T^{2} \)
43 \( 1 + 2.75iT - 43T^{2} \)
47 \( 1 - 1.36iT - 47T^{2} \)
53 \( 1 + 4.45iT - 53T^{2} \)
59 \( 1 - 5.67T + 59T^{2} \)
61 \( 1 + 2.35iT - 61T^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 + 4.93iT - 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 + 7.44iT - 79T^{2} \)
83 \( 1 - 9.94iT - 83T^{2} \)
89 \( 1 - 1.94T + 89T^{2} \)
97 \( 1 - 7.31T + 97T^{2} \)
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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

−9.507275899346181167072907763284, −9.100905493583890506709092670876, −8.244238237795706359770210399088, −7.45081707106556036193291147812, −6.96170763515691532110485754908, −5.48447447492601905800472030854, −4.65263774233056219722045289458, −3.98121356031656348623885656465, −2.47050931356630234373368242223, −1.54576230917631787977254073052, 1.28535262006922797793843464549, 2.66288607268351107539257268301, 3.16313395148615418007864960068, 3.81936823916714132051560083319, 5.43518504572116776845865848532, 6.18592521978801117529264073828, 7.68986681216966503218158459871, 8.056037774372817790215873712142, 8.851276988781979483029606191707, 9.674818255406348278187405567168

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