Properties

Label 2-1110-185.68-c1-0-8
Degree $2$
Conductor $1110$
Sign $0.673 - 0.739i$
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 + (−2.02 + 0.943i)5-s + (−0.707 + 0.707i)6-s + (−0.516 + 0.516i)7-s − 8-s − 1.00i·9-s + (2.02 − 0.943i)10-s − 0.497i·11-s + (0.707 − 0.707i)12-s − 1.53·13-s + (0.516 − 0.516i)14-s + (−0.766 + 2.10i)15-s + 16-s − 2.55i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.906 + 0.421i)5-s + (−0.288 + 0.288i)6-s + (−0.195 + 0.195i)7-s − 0.353·8-s − 0.333i·9-s + (0.641 − 0.298i)10-s − 0.149i·11-s + (0.204 − 0.204i)12-s − 0.424·13-s + (0.137 − 0.137i)14-s + (−0.197 + 0.542i)15-s + 0.250·16-s − 0.619i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.673 - 0.739i$
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: \((2,\ 1110,\ (\ :1/2),\ 0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9309930498\)
\(L(\frac12)\) \(\approx\) \(0.9309930498\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.02 - 0.943i)T \)
37 \( 1 + (-4.64 - 3.93i)T \)
good7 \( 1 + (0.516 - 0.516i)T - 7iT^{2} \)
11 \( 1 + 0.497iT - 11T^{2} \)
13 \( 1 + 1.53T + 13T^{2} \)
17 \( 1 + 2.55iT - 17T^{2} \)
19 \( 1 + (-2.36 - 2.36i)T + 19iT^{2} \)
23 \( 1 - 5.13T + 23T^{2} \)
29 \( 1 + (7.56 - 7.56i)T - 29iT^{2} \)
31 \( 1 + (-5.92 - 5.92i)T + 31iT^{2} \)
41 \( 1 + 2.70iT - 41T^{2} \)
43 \( 1 - 9.05T + 43T^{2} \)
47 \( 1 + (9.26 - 9.26i)T - 47iT^{2} \)
53 \( 1 + (0.413 + 0.413i)T + 53iT^{2} \)
59 \( 1 + (-5.91 - 5.91i)T + 59iT^{2} \)
61 \( 1 + (-10.7 - 10.7i)T + 61iT^{2} \)
67 \( 1 + (9.29 + 9.29i)T + 67iT^{2} \)
71 \( 1 + 4.10T + 71T^{2} \)
73 \( 1 + (2.63 - 2.63i)T - 73iT^{2} \)
79 \( 1 + (0.827 + 0.827i)T + 79iT^{2} \)
83 \( 1 + (-11.2 - 11.2i)T + 83iT^{2} \)
89 \( 1 + (-6.24 + 6.24i)T - 89iT^{2} \)
97 \( 1 + 3.13iT - 97T^{2} \)
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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

−9.797218479061382115580196191071, −9.039265610485041319837603387943, −8.312359040168751117151585211522, −7.38234287805761258737240990393, −7.09193634996875701363658910395, −5.96631397697178728713020018795, −4.71588185876787371066370106118, −3.34763853898715103724798971643, −2.72368081860261818147910316792, −1.12233743422713703299977274235, 0.58577271535074833162145820315, 2.28664987087633750713533077299, 3.48872665675930947327918422527, 4.34429121447940162389163787307, 5.38138005674280262683165644484, 6.64164902839562000084005373443, 7.60546686565475214159441527142, 8.031845217046993916120501545079, 9.020680237446006550752720687563, 9.569689180540617294233647752665

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