↑

Properties

Label 2-1560-1560.1403-c0-0-18
Degree $2$
Conductor $1560$
Sign $0.229 + 0.973i$
Analytic cond. $0.778541$
Root an. cond. $0.882349$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + 3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)6-s + (0.707 − 0.707i)8-s + 9-s − 1.00·10-s + 1.00i·12-s + (−0.707 − 0.707i)13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (−1 − i)17-s + (−0.707 − 0.707i)18-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 3-s + 1.00i·4-s + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)6-s + (0.707 − 0.707i)8-s + 9-s − 1.00·10-s + 1.00i·12-s + (−0.707 − 0.707i)13-s + (0.707 − 0.707i)15-s − 1.00·16-s + (−1 − i)17-s + (−0.707 − 0.707i)18-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.778541\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.207295309\)
\(L(\frac12)\) \(\approx\) \(1.207295309\)
\(L(1)\) 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 \( 1 - T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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

−9.525255445162931674694316108853, −8.751433679869942619863490731054, −8.207348731838441561691465002770, −7.36285726958908757256304145095, −6.49806366025205173377229351045, −4.98146631992474167106096851895, −4.33158477315213165976867180788, −2.96753746262196121044430277518, −2.39858619384828046134713780174, −1.19233075462391775007203047007, 1.81470004441514131644055188475, 2.43682870814301258940203836563, 3.87653230167456957326016644601, 4.91908464627980076951753285388, 6.06013572796187894318966208693, 6.83189348450268094654289111670, 7.32227883268061920976117464011, 8.319503307016658190088215528019, 8.992962688651507504171737095200, 9.565337438382639171681223511974

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