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Properties

Label 2-1110-185.68-c1-0-7
Degree $2$
Conductor $1110$
Sign $0.763 - 0.645i$
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 + (−1.65 − 1.50i)5-s + (−0.707 + 0.707i)6-s + (−1.06 + 1.06i)7-s − 8-s − 1.00i·9-s + (1.65 + 1.50i)10-s + 4.34i·11-s + (0.707 − 0.707i)12-s + 4.56·13-s + (1.06 − 1.06i)14-s + (−2.23 + 0.111i)15-s + 16-s + 3.41i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.741 − 0.670i)5-s + (−0.288 + 0.288i)6-s + (−0.401 + 0.401i)7-s − 0.353·8-s − 0.333i·9-s + (0.524 + 0.474i)10-s + 1.31i·11-s + (0.204 − 0.204i)12-s + 1.26·13-s + (0.283 − 0.283i)14-s + (−0.576 + 0.0287i)15-s + 0.250·16-s + 0.827i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.763 - 0.645i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.763 - 0.645i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9509441603\)
\(L(\frac12)\) \(\approx\) \(0.9509441603\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 1.50i)T \)
37 \( 1 + (-5.73 + 2.01i)T \)
good7 \( 1 + (1.06 - 1.06i)T - 7iT^{2} \)
11 \( 1 - 4.34iT - 11T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 - 3.41iT - 17T^{2} \)
19 \( 1 + (3.26 + 3.26i)T + 19iT^{2} \)
23 \( 1 + 3.95T + 23T^{2} \)
29 \( 1 + (1.85 - 1.85i)T - 29iT^{2} \)
31 \( 1 + (-3.73 - 3.73i)T + 31iT^{2} \)
41 \( 1 - 12.2iT - 41T^{2} \)
43 \( 1 - 2.82T + 43T^{2} \)
47 \( 1 + (-5.10 + 5.10i)T - 47iT^{2} \)
53 \( 1 + (-4.58 - 4.58i)T + 53iT^{2} \)
59 \( 1 + (2.79 + 2.79i)T + 59iT^{2} \)
61 \( 1 + (-1.42 - 1.42i)T + 61iT^{2} \)
67 \( 1 + (-3.17 - 3.17i)T + 67iT^{2} \)
71 \( 1 - 9.34T + 71T^{2} \)
73 \( 1 + (-6.35 + 6.35i)T - 73iT^{2} \)
79 \( 1 + (0.205 + 0.205i)T + 79iT^{2} \)
83 \( 1 + (2.55 + 2.55i)T + 83iT^{2} \)
89 \( 1 + (4.88 - 4.88i)T - 89iT^{2} \)
97 \( 1 - 7.85iT - 97T^{2} \)
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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

−9.688091913885006237750649972149, −8.977396155057647523014090743650, −8.342837980931647337273626980932, −7.70561374168446646211958450152, −6.74316400237666936246409113459, −5.99063840503991987172930822825, −4.58172402248178208215037282194, −3.67903880320700547508617341134, −2.35668136826758427322990654664, −1.19707484556472575997215626162, 0.57374140252667364507676565116, 2.45955206608280380804777672734, 3.55360830400826513920734548871, 4.03356000362893613311311806688, 5.80181809911102081530089707159, 6.46736645371536831377037399408, 7.50759612280480225964121745828, 8.237236646604684268231299370106, 8.773620454815563114910461043278, 9.818465019942533613450471556181

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