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Properties

Label 2-1110-185.43-c1-0-10
Degree $2$
Conductor $1110$
Sign $-0.697 - 0.716i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
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  + i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.0221 + 2.23i)5-s + (0.707 + 0.707i)6-s + (−0.281 + 0.281i)7-s − i·8-s − 1.00i·9-s + (−2.23 − 0.0221i)10-s + 1.64i·11-s + (−0.707 + 0.707i)12-s + 3.15i·13-s + (−0.281 − 0.281i)14-s + (1.56 + 1.59i)15-s + 16-s + 3.44·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (−0.00990 + 0.999i)5-s + (0.288 + 0.288i)6-s + (−0.106 + 0.106i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.707 − 0.00700i)10-s + 0.496i·11-s + (−0.204 + 0.204i)12-s + 0.875i·13-s + (−0.0752 − 0.0752i)14-s + (0.404 + 0.412i)15-s + 0.250·16-s + 0.835·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.697 - 0.716i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.697 - 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.453094908\)
\(L(\frac12)\) \(\approx\) \(1.453094908\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.0221 - 2.23i)T \)
37 \( 1 + (6.01 + 0.925i)T \)
good7 \( 1 + (0.281 - 0.281i)T - 7iT^{2} \)
11 \( 1 - 1.64iT - 11T^{2} \)
13 \( 1 - 3.15iT - 13T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 + (-1.85 + 1.85i)T - 19iT^{2} \)
23 \( 1 - 3.91iT - 23T^{2} \)
29 \( 1 + (2.67 + 2.67i)T + 29iT^{2} \)
31 \( 1 + (6.77 - 6.77i)T - 31iT^{2} \)
41 \( 1 - 8.72iT - 41T^{2} \)
43 \( 1 - 1.27iT - 43T^{2} \)
47 \( 1 + (0.752 - 0.752i)T - 47iT^{2} \)
53 \( 1 + (-3.88 - 3.88i)T + 53iT^{2} \)
59 \( 1 + (-6.53 + 6.53i)T - 59iT^{2} \)
61 \( 1 + (0.144 - 0.144i)T - 61iT^{2} \)
67 \( 1 + (-0.492 - 0.492i)T + 67iT^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + (0.804 - 0.804i)T - 73iT^{2} \)
79 \( 1 + (2.86 - 2.86i)T - 79iT^{2} \)
83 \( 1 + (-3.45 - 3.45i)T + 83iT^{2} \)
89 \( 1 + (12.5 + 12.5i)T + 89iT^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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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

−9.815456848007737881798082303537, −9.378163176753709500244054600158, −8.332685481237964368577148469209, −7.37726410083307706485689111214, −7.05532851463685871701104748724, −6.14554085924327870379396838886, −5.19702673956171940730865997140, −3.90986612248543767134545444417, −3.03118668663650704533218169908, −1.70711366816363643101787780221, 0.61784406643691498223982933384, 2.02142117873643011054052773904, 3.37125549158279992889274226759, 3.98326946142540303008510784223, 5.24785125874958343780928248585, 5.65384298326321565491784272163, 7.31810416572421159102472745423, 8.236033898555009726887643095932, 8.762513968227471746814463910493, 9.628441824610487132149855653686

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