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Properties

Label 2-1560-1560.467-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  + (0.707 − 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: $\chi_{1560} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :0),\ -0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.949461353\)
\(L(\frac12)\) \(\approx\) \(1.949461353\)
\(L(1)\) 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 \( 1 - iT^{2} \)
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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

−9.318254180093587696794425966298, −8.693544513410649875344856613537, −8.036665534674352035444121856644, −7.01842154701136052638403789465, −5.97859421161292538745308059314, −4.94805255767929023489199281854, −4.03450169591669412052882930906, −3.57578624758234445644758624542, −2.40121690528520517899430977326, −1.25763938007383682012569400836, 2.26622689572498962660598281407, 3.12210891156062601440413911045, 4.03957937575070272124451128453, 4.54211115151797703555642602882, 5.95858762495327392133573555824, 6.82638918579746069196331755961, 7.40566923219575296581799754904, 8.019237895641250556750984560259, 8.992615175265403683255255833536, 9.405666401189320818258847350850

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