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Properties

Label 2-1110-185.68-c1-0-13
Degree $2$
Conductor $1110$
Sign $0.812 - 0.583i$
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 + (−2.22 + 0.220i)5-s + (0.707 − 0.707i)6-s + (−1.18 + 1.18i)7-s + 8-s − 1.00i·9-s + (−2.22 + 0.220i)10-s + 4.79i·11-s + (0.707 − 0.707i)12-s + 4.44·13-s + (−1.18 + 1.18i)14-s + (−1.41 + 1.72i)15-s + 16-s + 1.30i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.995 + 0.0987i)5-s + (0.288 − 0.288i)6-s + (−0.448 + 0.448i)7-s + 0.353·8-s − 0.333i·9-s + (−0.703 + 0.0698i)10-s + 1.44i·11-s + (0.204 − 0.204i)12-s + 1.23·13-s + (−0.316 + 0.316i)14-s + (−0.365 + 0.446i)15-s + 0.250·16-s + 0.315i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.812 - 0.583i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.812 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.386417949\)
\(L(\frac12)\) \(\approx\) \(2.386417949\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 - 0.220i)T \)
37 \( 1 + (5.10 + 3.31i)T \)
good7 \( 1 + (1.18 - 1.18i)T - 7iT^{2} \)
11 \( 1 - 4.79iT - 11T^{2} \)
13 \( 1 - 4.44T + 13T^{2} \)
17 \( 1 - 1.30iT - 17T^{2} \)
19 \( 1 + (-4.46 - 4.46i)T + 19iT^{2} \)
23 \( 1 + 0.933T + 23T^{2} \)
29 \( 1 + (-3.66 + 3.66i)T - 29iT^{2} \)
31 \( 1 + (-4.54 - 4.54i)T + 31iT^{2} \)
41 \( 1 + 7.79iT - 41T^{2} \)
43 \( 1 - 2.02T + 43T^{2} \)
47 \( 1 + (6.81 - 6.81i)T - 47iT^{2} \)
53 \( 1 + (-5.08 - 5.08i)T + 53iT^{2} \)
59 \( 1 + (-3.87 - 3.87i)T + 59iT^{2} \)
61 \( 1 + (6.36 + 6.36i)T + 61iT^{2} \)
67 \( 1 + (0.639 + 0.639i)T + 67iT^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 + (-1.06 + 1.06i)T - 73iT^{2} \)
79 \( 1 + (-2.21 - 2.21i)T + 79iT^{2} \)
83 \( 1 + (6.32 + 6.32i)T + 83iT^{2} \)
89 \( 1 + (3.11 - 3.11i)T - 89iT^{2} \)
97 \( 1 + 0.0235iT - 97T^{2} \)
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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

−10.00853154078045075363692774365, −8.966879812022747877709678238897, −8.094116611269768992681768588748, −7.40256322863112182616524779544, −6.60590951792318709362816571698, −5.73929999561332765498442037970, −4.50835032421155614331805328992, −3.71303798845757439262254229276, −2.85245885038729717397868693337, −1.50414972042397588818383444782, 0.886188685250004521964924785743, 3.06145213869906322819166352316, 3.41330649728366092338201480933, 4.36163596855276284814146445513, 5.31865008375661415938202478974, 6.39579119963730650722039807609, 7.18199395855025248878221075281, 8.271721093768180164303214978134, 8.675506978742763295125515090850, 9.848457527672044406864805444026

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