↑

Properties

Label 2-1190-17.13-c1-0-16
Degree $2$
Conductor $1190$
Sign $0.832 - 0.554i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (2.23 + 2.23i)3-s − 4-s + (−0.707 − 0.707i)5-s + (2.23 − 2.23i)6-s + (−0.707 + 0.707i)7-s + i·8-s + 6.98i·9-s + (−0.707 + 0.707i)10-s + (2.20 − 2.20i)11-s + (−2.23 − 2.23i)12-s + 5.45·13-s + (0.707 + 0.707i)14-s − 3.16i·15-s + 16-s + (1.29 − 3.91i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (1.29 + 1.29i)3-s − 0.5·4-s + (−0.316 − 0.316i)5-s + (0.912 − 0.912i)6-s + (−0.267 + 0.267i)7-s + 0.353i·8-s + 2.32i·9-s + (−0.223 + 0.223i)10-s + (0.663 − 0.663i)11-s + (−0.645 − 0.645i)12-s + 1.51·13-s + (0.188 + 0.188i)14-s − 0.816i·15-s + 0.250·16-s + (0.314 − 0.949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1190\)    =    \(2 \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.832 - 0.554i$
Analytic conductor: \(9.50219\)
Root analytic conductor: \(3.08256\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1190} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1190,\ (\ :1/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.400189374\)
\(L(\frac12)\) \(\approx\) \(2.400189374\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-1.29 + 3.91i)T \)
good3 \( 1 + (-2.23 - 2.23i)T + 3iT^{2} \)
11 \( 1 + (-2.20 + 2.20i)T - 11iT^{2} \)
13 \( 1 - 5.45T + 13T^{2} \)
19 \( 1 - 3.06iT - 19T^{2} \)
23 \( 1 + (2.89 - 2.89i)T - 23iT^{2} \)
29 \( 1 + (-0.0545 - 0.0545i)T + 29iT^{2} \)
31 \( 1 + (-5.06 - 5.06i)T + 31iT^{2} \)
37 \( 1 + (-6.45 - 6.45i)T + 37iT^{2} \)
41 \( 1 + (8.44 - 8.44i)T - 41iT^{2} \)
43 \( 1 + 2.97iT - 43T^{2} \)
47 \( 1 + 6.50T + 47T^{2} \)
53 \( 1 - 6.35iT - 53T^{2} \)
59 \( 1 + 7.19iT - 59T^{2} \)
61 \( 1 + (-6.40 + 6.40i)T - 61iT^{2} \)
67 \( 1 - 7.35T + 67T^{2} \)
71 \( 1 + (2.91 + 2.91i)T + 71iT^{2} \)
73 \( 1 + (6.10 + 6.10i)T + 73iT^{2} \)
79 \( 1 + (-5.77 + 5.77i)T - 79iT^{2} \)
83 \( 1 - 1.84iT - 83T^{2} \)
89 \( 1 - 2.43T + 89T^{2} \)
97 \( 1 + (6.80 + 6.80i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.714992252199929087045134113995, −9.180257071984587823045599532459, −8.324824163974691159706562825590, −8.095444043443983901571449869733, −6.37536143614454347992056159966, −5.20283883449915467328332731498, −4.29230424002994341789957602036, −3.44576331979399049622482753805, −3.06539353255732520697191800416, −1.49230560072331785627609904522, 1.03630137682139613326835262511, 2.28258553686024004487393287011, 3.57383601920084108656200320363, 4.12490033695581686189620484393, 5.98870780782824037369147129711, 6.60924498563951996150194709399, 7.18185679863773769576890341882, 8.107587353297472820688835988715, 8.488166561650041710612727924270, 9.316710004818062296931834384209

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