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Properties

Label 2-1110-111.68-c1-0-42
Degree $2$
Conductor $1110$
Sign $-0.879 + 0.475i$
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  + (−0.707 + 0.707i)2-s + (1.64 − 0.535i)3-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (−0.786 + 1.54i)6-s − 2.53·7-s + (0.707 + 0.707i)8-s + (2.42 − 1.76i)9-s + 1.00·10-s − 1.55·11-s + (−0.535 − 1.64i)12-s + (−1.11 + 1.11i)13-s + (1.79 − 1.79i)14-s + (−1.54 − 0.786i)15-s − 1.00·16-s + (−4.84 − 4.84i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.951 − 0.309i)3-s − 0.500i·4-s + (−0.316 − 0.316i)5-s + (−0.320 + 0.630i)6-s − 0.959·7-s + (0.250 + 0.250i)8-s + (0.808 − 0.587i)9-s + 0.316·10-s − 0.468·11-s + (−0.154 − 0.475i)12-s + (−0.309 + 0.309i)13-s + (0.479 − 0.479i)14-s + (−0.398 − 0.203i)15-s − 0.250·16-s + (−1.17 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.879 + 0.475i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3336785348\)
\(L(\frac12)\) \(\approx\) \(0.3336785348\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-1.64 + 0.535i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-3.85 + 4.70i)T \)
good7 \( 1 + 2.53T + 7T^{2} \)
11 \( 1 + 1.55T + 11T^{2} \)
13 \( 1 + (1.11 - 1.11i)T - 13iT^{2} \)
17 \( 1 + (4.84 + 4.84i)T + 17iT^{2} \)
19 \( 1 + (2.49 - 2.49i)T - 19iT^{2} \)
23 \( 1 + (0.856 + 0.856i)T + 23iT^{2} \)
29 \( 1 + (5.19 - 5.19i)T - 29iT^{2} \)
31 \( 1 + (-1.70 - 1.70i)T + 31iT^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + (2.78 - 2.78i)T - 43iT^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 - 5.16iT - 53T^{2} \)
59 \( 1 + (8.36 + 8.36i)T + 59iT^{2} \)
61 \( 1 + (-7.41 - 7.41i)T + 61iT^{2} \)
67 \( 1 + 1.98iT - 67T^{2} \)
71 \( 1 + 3.37iT - 71T^{2} \)
73 \( 1 - 0.480iT - 73T^{2} \)
79 \( 1 + (-6.52 + 6.52i)T - 79iT^{2} \)
83 \( 1 - 8.64iT - 83T^{2} \)
89 \( 1 + (10.5 - 10.5i)T - 89iT^{2} \)
97 \( 1 + (5.01 - 5.01i)T - 97iT^{2} \)
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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

−9.324982984702778114119260879837, −8.687659625622054457522163580255, −7.925088496562756710851426616021, −7.00121497189273694718195475822, −6.58701506585689783567253104249, −5.23829055840093331055738303059, −4.15901200202830434584846815860, −3.04737345174291295424707916363, −1.92774332203192498284116359238, −0.14049134421063047807453264090, 2.03216175943198025036201372870, 2.90445995029745805650206508421, 3.76598793940258117098138835728, 4.63142840552106814281170032600, 6.21568808266862811779418908028, 7.06625655936887120146562149097, 8.010431011651040545913360099503, 8.564704583279415046806756725760, 9.492711861726393876953205045636, 10.05341952275112579565776485269

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