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Properties

Label 2-111-37.23-c2-0-3
Degree $2$
Conductor $111$
Sign $-0.769 - 0.638i$
Analytic cond. $3.02453$
Root an. cond. $1.73911$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.821 + 3.06i)2-s + (1.5 − 0.866i)3-s + (−5.26 + 3.03i)4-s + (−1.49 + 5.59i)5-s + (3.88 + 3.88i)6-s + (−0.749 − 1.29i)7-s + (−4.66 − 4.66i)8-s + (1.5 − 2.59i)9-s − 18.3·10-s + 7.11i·11-s + (−5.26 + 9.11i)12-s + (0.492 − 1.83i)13-s + (3.36 − 3.36i)14-s + (2.59 + 9.68i)15-s + (−1.68 + 2.91i)16-s + (18.7 − 5.02i)17-s + ⋯
L(s)  = 1  + (0.410 + 1.53i)2-s + (0.5 − 0.288i)3-s + (−1.31 + 0.759i)4-s + (−0.299 + 1.11i)5-s + (0.648 + 0.648i)6-s + (−0.107 − 0.185i)7-s + (−0.583 − 0.583i)8-s + (0.166 − 0.288i)9-s − 1.83·10-s + 0.647i·11-s + (−0.438 + 0.759i)12-s + (0.0378 − 0.141i)13-s + (0.240 − 0.240i)14-s + (0.173 + 0.645i)15-s + (−0.105 + 0.182i)16-s + (1.10 − 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.769 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.769 - 0.638i$
Analytic conductor: \(3.02453\)
Root analytic conductor: \(1.73911\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 111,\ (\ :1),\ -0.769 - 0.638i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.596073 + 1.65327i\)
\(L(\frac12)\) \(\approx\) \(0.596073 + 1.65327i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
37 \( 1 + (-29.2 - 22.6i)T \)
good2 \( 1 + (-0.821 - 3.06i)T + (-3.46 + 2i)T^{2} \)
5 \( 1 + (1.49 - 5.59i)T + (-21.6 - 12.5i)T^{2} \)
7 \( 1 + (0.749 + 1.29i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 - 7.11iT - 121T^{2} \)
13 \( 1 + (-0.492 + 1.83i)T + (-146. - 84.5i)T^{2} \)
17 \( 1 + (-18.7 + 5.02i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-7.53 + 28.1i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-2.29 - 2.29i)T + 529iT^{2} \)
29 \( 1 + (-33.6 + 33.6i)T - 841iT^{2} \)
31 \( 1 + (21.4 - 21.4i)T - 961iT^{2} \)
41 \( 1 + (-0.311 + 0.179i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (3.53 + 3.53i)T + 1.84e3iT^{2} \)
47 \( 1 - 5.35T + 2.20e3T^{2} \)
53 \( 1 + (25.0 - 43.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (73.4 - 19.6i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (55.8 + 14.9i)T + (3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (80.5 - 46.4i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (37.8 + 65.6i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 + (-10.1 + 37.8i)T + (-5.40e3 - 3.12e3i)T^{2} \)
83 \( 1 + (-69.7 + 120. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-27.8 - 104. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (124. + 124. i)T + 9.40e3iT^{2} \)
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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

−14.03660634424246592184526047609, −13.35371206390027146158859908864, −11.97641559234644998477673160610, −10.54484731620861256774131337277, −9.183149940748976002257833601521, −7.75961850119777320437171511619, −7.21500796778854333281732762897, −6.26591091926395922712119187269, −4.70034862438597366717557692155, −3.07223018725916982962933601005, 1.29993883156514037777368812764, 3.16960215277519055033303986318, 4.25364194591101399537437321366, 5.52976630168780500320889304453, 7.946742764441893949174721497799, 9.008650379354133391065163170399, 9.944362927628240096570296820395, 10.99131227688921799529669506716, 12.24313108525889419967039706160, 12.59389926615732741926289273952

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