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Properties

Label 2-880-5.4-c1-0-18
Degree $2$
Conductor $880$
Sign $0.447 + 0.894i$
Analytic cond. $7.02683$
Root an. cond. $2.65081$
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  − 2i·3-s + (1 + 2i)5-s − 9-s − 11-s − 2i·13-s + (4 − 2i)15-s − 6i·17-s + 4·19-s + 2i·23-s + (−3 + 4i)25-s − 4i·27-s + 10·29-s + 8·31-s + 2i·33-s − 8i·37-s + ⋯
L(s)  = 1  − 1.15i·3-s + (0.447 + 0.894i)5-s − 0.333·9-s − 0.301·11-s − 0.554i·13-s + (1.03 − 0.516i)15-s − 1.45i·17-s + 0.917·19-s + 0.417i·23-s + (−0.600 + 0.800i)25-s − 0.769i·27-s + 1.85·29-s + 1.43·31-s + 0.348i·33-s − 1.31i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47015 - 0.908606i\)
\(L(\frac12)\) \(\approx\) \(1.47015 - 0.908606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2i)T \)
11 \( 1 + T \)
good3 \( 1 + 2iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−10.04872535952889784135943192477, −9.221054819882628306815200663172, −7.938341896180966238230761038117, −7.41059750890826702927488953032, −6.67014910592668976950920544130, −5.87880722865103928890246395907, −4.80714170667068985949446140034, −3.12417489853236463576232772935, −2.42143222918674025364408856994, −0.968851563305921845223404335686, 1.38999656468989766897772259345, 2.98591313598953536954388675937, 4.29615214904181892139254574304, 4.73080260807084545080423119924, 5.75397678963261367959444872637, 6.67827126616354226654120085948, 8.139149063667877527083079251091, 8.665857900705018395500473567388, 9.637033391259177570664430833094, 10.12158826123051551981185885567

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