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Properties

Label 2-560-7.2-c1-0-4
Degree $2$
Conductor $560$
Sign $0.991 + 0.126i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)3-s + (−0.5 + 0.866i)5-s + (2 + 1.73i)7-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s − 13-s + 1.99·15-s + (3 + 5.19i)17-s + (−0.5 + 0.866i)19-s + (0.999 − 5.19i)21-s + (4.5 − 7.79i)23-s + (−0.499 − 0.866i)25-s − 4.00·27-s + 6·29-s + (4 + 6.92i)31-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (−0.223 + 0.387i)5-s + (0.755 + 0.654i)7-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s − 0.277·13-s + 0.516·15-s + (0.727 + 1.26i)17-s + (−0.114 + 0.198i)19-s + (0.218 − 1.13i)21-s + (0.938 − 1.62i)23-s + (−0.0999 − 0.173i)25-s − 0.769·27-s + 1.11·29-s + (0.718 + 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29259 - 0.0820282i\)
\(L(\frac12)\) \(\approx\) \(1.29259 - 0.0820282i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2 - 1.73i)T \)
good3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.5 + 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.86704440477331386789536550153, −10.06338745012080907248972395109, −8.745984057308770551915208756826, −8.007447667696574132210763924403, −6.98392003663285543211231151675, −6.39136855850539391791999376875, −5.31767343523139848179766350979, −4.17693115187215572523974092457, −2.51229013654512687151908651352, −1.28472033341831520904129462361, 1.01367815226189408960029047685, 3.18066882536543965000805940961, 4.40997271002356585730612299734, 4.94932386074771125384826692131, 5.94236139111281418627064752854, 7.33579708634229134842377724061, 8.072358133460878407128521996081, 9.313362275315100698859506168027, 9.852905168947020426447899319820, 10.99359032420947259967469325883

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