Properties

Label 2-1110-185.117-c1-0-13
Degree $2$
Conductor $1110$
Sign $0.935 - 0.352i$
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  − 2-s + (0.707 + 0.707i)3-s + 4-s + (−1.64 − 1.51i)5-s + (−0.707 − 0.707i)6-s + (2.75 + 2.75i)7-s − 8-s + 1.00i·9-s + (1.64 + 1.51i)10-s − 0.350i·11-s + (0.707 + 0.707i)12-s + 3.27·13-s + (−2.75 − 2.75i)14-s + (−0.0966 − 2.23i)15-s + 16-s − 7.07i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (−0.737 − 0.675i)5-s + (−0.288 − 0.288i)6-s + (1.04 + 1.04i)7-s − 0.353·8-s + 0.333i·9-s + (0.521 + 0.477i)10-s − 0.105i·11-s + (0.204 + 0.204i)12-s + 0.907·13-s + (−0.737 − 0.737i)14-s + (−0.0249 − 0.576i)15-s + 0.250·16-s − 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.935 - 0.352i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.935 - 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.388918135\)
\(L(\frac12)\) \(\approx\) \(1.388918135\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (1.64 + 1.51i)T \)
37 \( 1 + (3.58 + 4.91i)T \)
good7 \( 1 + (-2.75 - 2.75i)T + 7iT^{2} \)
11 \( 1 + 0.350iT - 11T^{2} \)
13 \( 1 - 3.27T + 13T^{2} \)
17 \( 1 + 7.07iT - 17T^{2} \)
19 \( 1 + (-2.01 + 2.01i)T - 19iT^{2} \)
23 \( 1 + 1.24T + 23T^{2} \)
29 \( 1 + (-6.14 - 6.14i)T + 29iT^{2} \)
31 \( 1 + (2.76 - 2.76i)T - 31iT^{2} \)
41 \( 1 - 5.57iT - 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + (-5.76 - 5.76i)T + 47iT^{2} \)
53 \( 1 + (3.45 - 3.45i)T - 53iT^{2} \)
59 \( 1 + (-10.6 + 10.6i)T - 59iT^{2} \)
61 \( 1 + (-1.30 + 1.30i)T - 61iT^{2} \)
67 \( 1 + (-10.6 + 10.6i)T - 67iT^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 + (-0.998 - 0.998i)T + 73iT^{2} \)
79 \( 1 + (-2.47 + 2.47i)T - 79iT^{2} \)
83 \( 1 + (10.6 - 10.6i)T - 83iT^{2} \)
89 \( 1 + (-3.03 - 3.03i)T + 89iT^{2} \)
97 \( 1 - 7.94iT - 97T^{2} \)
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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

−9.469134366393036335586733273378, −9.044173539903431537151847804689, −8.360787997459584819865953152209, −7.80514187074968380110209268526, −6.79081814829835070451903360747, −5.33088661444227240412864193744, −4.89503427232038167874981452448, −3.55746438986742960951815939167, −2.45763796710747652149893370318, −1.07312420925853396853198351342, 1.00493091243264479846170757151, 2.13941173763256503041873268226, 3.63413813436463853562615600101, 4.16641939195127968511719185452, 5.83464438597716687909234607964, 6.79399338241964978614796363575, 7.48273267723817065135198618751, 8.291173708759761342700917747488, 8.458622496867004947228046066178, 10.04394834146959566117911750082

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