Properties

Label 2-1110-185.117-c1-0-20
Degree $2$
Conductor $1110$
Sign $0.855 - 0.517i$
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 + (1.91 − 1.16i)5-s + (−0.707 − 0.707i)6-s + (2.90 + 2.90i)7-s − 8-s + 1.00i·9-s + (−1.91 + 1.16i)10-s − 2.74i·11-s + (0.707 + 0.707i)12-s + 5.21·13-s + (−2.90 − 2.90i)14-s + (2.17 + 0.529i)15-s + 16-s + 4.16i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (0.854 − 0.519i)5-s + (−0.288 − 0.288i)6-s + (1.09 + 1.09i)7-s − 0.353·8-s + 0.333i·9-s + (−0.604 + 0.367i)10-s − 0.828i·11-s + (0.204 + 0.204i)12-s + 1.44·13-s + (−0.776 − 0.776i)14-s + (0.560 + 0.136i)15-s + 0.250·16-s + 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.855 - 0.517i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.855 - 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.875479123\)
\(L(\frac12)\) \(\approx\) \(1.875479123\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 + 1.16i)T \)
37 \( 1 + (2.77 - 5.41i)T \)
good7 \( 1 + (-2.90 - 2.90i)T + 7iT^{2} \)
11 \( 1 + 2.74iT - 11T^{2} \)
13 \( 1 - 5.21T + 13T^{2} \)
17 \( 1 - 4.16iT - 17T^{2} \)
19 \( 1 + (1.87 - 1.87i)T - 19iT^{2} \)
23 \( 1 + 0.941T + 23T^{2} \)
29 \( 1 + (4.28 + 4.28i)T + 29iT^{2} \)
31 \( 1 + (-5.51 + 5.51i)T - 31iT^{2} \)
41 \( 1 - 7.59iT - 41T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 + (6.04 + 6.04i)T + 47iT^{2} \)
53 \( 1 + (-8.31 + 8.31i)T - 53iT^{2} \)
59 \( 1 + (-1.92 + 1.92i)T - 59iT^{2} \)
61 \( 1 + (8.68 - 8.68i)T - 61iT^{2} \)
67 \( 1 + (-5.51 + 5.51i)T - 67iT^{2} \)
71 \( 1 + 8.51T + 71T^{2} \)
73 \( 1 + (2.98 + 2.98i)T + 73iT^{2} \)
79 \( 1 + (-1.46 + 1.46i)T - 79iT^{2} \)
83 \( 1 + (-7.26 + 7.26i)T - 83iT^{2} \)
89 \( 1 + (1.62 + 1.62i)T + 89iT^{2} \)
97 \( 1 - 10.0iT - 97T^{2} \)
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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

−9.823735311340896840269120604491, −8.867902761405378241049626254541, −8.363896163479852621506546837793, −8.117637373548134910013606481456, −6.22285496490021945227442126920, −5.90920557824539407914007744486, −4.83780676476488238027919007310, −3.58562852436455093476790104674, −2.22956278319595333362828390009, −1.43629437395867601940618130461, 1.21219156266204193682351677567, 2.00174328133574668567988204381, 3.27737854161305339166040164428, 4.52262744009472317071675906824, 5.68791063203019512070214283999, 6.93336677798129763671327660714, 7.12139607633051008158832667608, 8.175538292797531323130344939751, 8.913728018210353981312240224661, 9.730750994570100104204367056472

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