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Properties

Label 2-1560-1560.1403-c0-0-17
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)2-s + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)4-s + (0.923 + 0.382i)5-s + (−0.923 − 0.382i)6-s + (0.382 − 0.923i)8-s + 1.00i·9-s + 10-s − 0.765·11-s − 12-s + (0.707 − 0.707i)13-s + (−0.382 − 0.923i)15-s − i·16-s + (0.382 + 0.923i)18-s + (0.923 − 0.382i)20-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)4-s + (0.923 + 0.382i)5-s + (−0.923 − 0.382i)6-s + (0.382 − 0.923i)8-s + 1.00i·9-s + 10-s − 0.765·11-s − 12-s + (0.707 − 0.707i)13-s + (−0.382 − 0.923i)15-s − i·16-s + (0.382 + 0.923i)18-s + (0.923 − 0.382i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.771971157\)
\(L(\frac12)\) \(\approx\) \(1.771971157\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + 0.765T + T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 1.84T + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 1.84iT - T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 0.765iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
89 \( 1 - 0.765iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.910050262615561939861646363434, −8.606258374700528102377797192023, −7.51438115367535812145682634824, −6.80379503062807381635410784481, −5.93770493961255733591015025030, −5.54620426870470469832882333026, −4.67804794510023469226798853657, −3.25363107092413021471558226961, −2.36763761140940274544485370535, −1.32573228339451575568794014122, 1.80732844300405348404991807916, 3.13590189295346853960459546997, 4.14203036302488975250800946887, 5.02932300600768411978431185198, 5.51663551428012382424857220621, 6.36533320991465972118048379281, 6.95113432790355447440706308222, 8.292284072281854212493352227975, 8.950260243622399241841978300250, 10.01311893890531292421213068726

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