↑

Properties

Label 2-1360-20.7-c1-0-9
Degree $2$
Conductor $1360$
Sign $0.787 - 0.616i$
Analytic cond. $10.8596$
Root an. cond. $3.29539$
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  + (−1.62 − 1.62i)3-s + (1.74 + 1.39i)5-s + (−1.20 + 1.20i)7-s + 2.26i·9-s + 2.64i·11-s + (3.80 − 3.80i)13-s + (−0.567 − 5.09i)15-s + (−0.707 − 0.707i)17-s − 4.76·19-s + 3.90·21-s + (0.848 + 0.848i)23-s + (1.09 + 4.87i)25-s + (−1.19 + 1.19i)27-s + 4.15i·29-s + 3.91i·31-s + ⋯
L(s)  = 1  + (−0.936 − 0.936i)3-s + (0.780 + 0.624i)5-s + (−0.454 + 0.454i)7-s + 0.755i·9-s + 0.796i·11-s + (1.05 − 1.05i)13-s + (−0.146 − 1.31i)15-s + (−0.171 − 0.171i)17-s − 1.09·19-s + 0.851·21-s + (0.176 + 0.176i)23-s + (0.219 + 0.975i)25-s + (−0.229 + 0.229i)27-s + 0.771i·29-s + 0.702i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1360\)    =    \(2^{4} \cdot 5 \cdot 17\)
Sign: $0.787 - 0.616i$
Analytic conductor: \(10.8596\)
Root analytic conductor: \(3.29539\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1360} (1327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1360,\ (\ :1/2),\ 0.787 - 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.131266910\)
\(L(\frac12)\) \(\approx\) \(1.131266910\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.74 - 1.39i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.62 + 1.62i)T + 3iT^{2} \)
7 \( 1 + (1.20 - 1.20i)T - 7iT^{2} \)
11 \( 1 - 2.64iT - 11T^{2} \)
13 \( 1 + (-3.80 + 3.80i)T - 13iT^{2} \)
19 \( 1 + 4.76T + 19T^{2} \)
23 \( 1 + (-0.848 - 0.848i)T + 23iT^{2} \)
29 \( 1 - 4.15iT - 29T^{2} \)
31 \( 1 - 3.91iT - 31T^{2} \)
37 \( 1 + (-4.47 - 4.47i)T + 37iT^{2} \)
41 \( 1 + 6.25T + 41T^{2} \)
43 \( 1 + (-7.59 - 7.59i)T + 43iT^{2} \)
47 \( 1 + (-6.80 + 6.80i)T - 47iT^{2} \)
53 \( 1 + (6.24 - 6.24i)T - 53iT^{2} \)
59 \( 1 - 2.42T + 59T^{2} \)
61 \( 1 + 0.875T + 61T^{2} \)
67 \( 1 + (-6.37 + 6.37i)T - 67iT^{2} \)
71 \( 1 + 3.01iT - 71T^{2} \)
73 \( 1 + (1.95 - 1.95i)T - 73iT^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 + (-12.7 - 12.7i)T + 83iT^{2} \)
89 \( 1 + 10.0iT - 89T^{2} \)
97 \( 1 + (-2.55 - 2.55i)T + 97iT^{2} \)
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

−9.754412567055200435607157143073, −8.936135337280423254068856404712, −7.88065028823642182344124718475, −6.94107526732636725444491636126, −6.35362797694349596708164091333, −5.84465078184684883738409783968, −4.94089853965058505482391150049, −3.40255923752307812784000141976, −2.32264424812669476419237836728, −1.19174354472445646957064204217, 0.58782666541627478453450826814, 2.14789252675236400653293248493, 3.85983807823724381267039443904, 4.32487087366843821431485443777, 5.40823202202693780363354739320, 6.12037951121231210403465062726, 6.60405706627655529690839279516, 8.111246299887560695345627346201, 8.983738853792726166926928051553, 9.519203997552688426975068649782

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