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Properties

Label 2-1110-185.142-c1-0-10
Degree $2$
Conductor $1110$
Sign $-0.841 - 0.539i$
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  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (−0.482 + 2.18i)5-s + (0.707 − 0.707i)6-s + (3.34 + 3.34i)7-s − i·8-s + 1.00i·9-s + (−2.18 − 0.482i)10-s + 0.666i·11-s + (0.707 + 0.707i)12-s + 2.34i·13-s + (−3.34 + 3.34i)14-s + (1.88 − 1.20i)15-s + 16-s + 2.53·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (−0.215 + 0.976i)5-s + (0.288 − 0.288i)6-s + (1.26 + 1.26i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.690 − 0.152i)10-s + 0.200i·11-s + (0.204 + 0.204i)12-s + 0.651i·13-s + (−0.893 + 0.893i)14-s + (0.486 − 0.310i)15-s + 0.250·16-s + 0.614·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $-0.841 - 0.539i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ -0.841 - 0.539i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.316474983\)
\(L(\frac12)\) \(\approx\) \(1.316474983\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.482 - 2.18i)T \)
37 \( 1 + (-3.34 - 5.08i)T \)
good7 \( 1 + (-3.34 - 3.34i)T + 7iT^{2} \)
11 \( 1 - 0.666iT - 11T^{2} \)
13 \( 1 - 2.34iT - 13T^{2} \)
17 \( 1 - 2.53T + 17T^{2} \)
19 \( 1 + (-2.61 - 2.61i)T + 19iT^{2} \)
23 \( 1 + 7.61iT - 23T^{2} \)
29 \( 1 + (1.83 - 1.83i)T - 29iT^{2} \)
31 \( 1 + (3.00 + 3.00i)T + 31iT^{2} \)
41 \( 1 - 6.19iT - 41T^{2} \)
43 \( 1 - 9.47iT - 43T^{2} \)
47 \( 1 + (5.52 + 5.52i)T + 47iT^{2} \)
53 \( 1 + (-3.00 + 3.00i)T - 53iT^{2} \)
59 \( 1 + (1.77 + 1.77i)T + 59iT^{2} \)
61 \( 1 + (1.33 + 1.33i)T + 61iT^{2} \)
67 \( 1 + (9.44 - 9.44i)T - 67iT^{2} \)
71 \( 1 - 3.44T + 71T^{2} \)
73 \( 1 + (7.93 + 7.93i)T + 73iT^{2} \)
79 \( 1 + (-8.95 - 8.95i)T + 79iT^{2} \)
83 \( 1 + (-6.18 + 6.18i)T - 83iT^{2} \)
89 \( 1 + (-0.703 + 0.703i)T - 89iT^{2} \)
97 \( 1 + 6.06T + 97T^{2} \)
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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

−10.15687534195436290793299069949, −9.217984124017694409217886856501, −8.155734392593831143916586284447, −7.81254345620736825429224650760, −6.75054139006675322154643123382, −6.07483593185062037116540501918, −5.23728156899641453337308495885, −4.35685050336042729224917844543, −2.85956868595864175008456280514, −1.69499315785437531865419848792, 0.66779454034346742265244984586, 1.59114371646485141482013612210, 3.47640342741405357588073956051, 4.18792729691567324864631965741, 5.11167882065933128129925016528, 5.57295684589659593779360525830, 7.40717254106099538596482474325, 7.81329071176847632785972689109, 8.867599487144752202325960860020, 9.604542359070245288690397872083

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