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Properties

Label 2-1110-185.68-c1-0-26
Degree $2$
Conductor $1110$
Sign $0.265 + 0.963i$
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 + (−2.23 + 0.132i)5-s + (−0.707 + 0.707i)6-s + (1.47 − 1.47i)7-s + 8-s − 1.00i·9-s + (−2.23 + 0.132i)10-s − 1.15i·11-s + (−0.707 + 0.707i)12-s − 2.88·13-s + (1.47 − 1.47i)14-s + (1.48 − 1.67i)15-s + 16-s − 5.62i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.408 + 0.408i)3-s + 0.5·4-s + (−0.998 + 0.0594i)5-s + (−0.288 + 0.288i)6-s + (0.559 − 0.559i)7-s + 0.353·8-s − 0.333i·9-s + (−0.705 + 0.0420i)10-s − 0.347i·11-s + (−0.204 + 0.204i)12-s − 0.801·13-s + (0.395 − 0.395i)14-s + (0.383 − 0.431i)15-s + 0.250·16-s − 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.265 + 0.963i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.265 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.476197360\)
\(L(\frac12)\) \(\approx\) \(1.476197360\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.132i)T \)
37 \( 1 + (4.72 - 3.82i)T \)
good7 \( 1 + (-1.47 + 1.47i)T - 7iT^{2} \)
11 \( 1 + 1.15iT - 11T^{2} \)
13 \( 1 + 2.88T + 13T^{2} \)
17 \( 1 + 5.62iT - 17T^{2} \)
19 \( 1 + (1.32 + 1.32i)T + 19iT^{2} \)
23 \( 1 + 2.70T + 23T^{2} \)
29 \( 1 + (-6.59 + 6.59i)T - 29iT^{2} \)
31 \( 1 + (0.920 + 0.920i)T + 31iT^{2} \)
41 \( 1 + 6.84iT - 41T^{2} \)
43 \( 1 - 8.92T + 43T^{2} \)
47 \( 1 + (-5.84 + 5.84i)T - 47iT^{2} \)
53 \( 1 + (4.27 + 4.27i)T + 53iT^{2} \)
59 \( 1 + (7.37 + 7.37i)T + 59iT^{2} \)
61 \( 1 + (-0.721 - 0.721i)T + 61iT^{2} \)
67 \( 1 + (4.32 + 4.32i)T + 67iT^{2} \)
71 \( 1 + 4.66T + 71T^{2} \)
73 \( 1 + (-0.815 + 0.815i)T - 73iT^{2} \)
79 \( 1 + (-11.8 - 11.8i)T + 79iT^{2} \)
83 \( 1 + (5.88 + 5.88i)T + 83iT^{2} \)
89 \( 1 + (-6.87 + 6.87i)T - 89iT^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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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

−9.904518977405003811728016737170, −8.791843383383621047456876607637, −7.75486787543166525308457559813, −7.21365880283378321114191210031, −6.23970443536716255275465190836, −5.02328539957857152403755181967, −4.54701287391310143452217685762, −3.66842101282518572076970701461, −2.53806850268489982274008452835, −0.55008698974440473537525124069, 1.55692517528979913384126221804, 2.78875427655260416657904724900, 4.07197466690661162691484157542, 4.77458292745090341913136426815, 5.69670989953110172563523567357, 6.60383490241372492169445997949, 7.51147781865070204220293568577, 8.118255219003298337663506262870, 9.002591380368514219653979219512, 10.43440321950853861636231165529

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