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Properties

Label 2-110-55.14-c1-0-4
Degree $2$
Conductor $110$
Sign $0.930 - 0.365i$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
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  + (0.587 + 0.809i)2-s + (1.53 − 0.5i)3-s + (−0.309 + 0.951i)4-s + (1.71 − 1.43i)5-s + (1.30 + 0.951i)6-s + (−4.76 − 1.54i)7-s + (−0.951 + 0.309i)8-s + (−0.309 + 0.224i)9-s + (2.16 + 0.549i)10-s + (−0.969 + 3.17i)11-s + 1.61i·12-s + (1.37 + 1.88i)13-s + (−1.54 − 4.76i)14-s + (1.92 − 3.06i)15-s + (−0.809 − 0.587i)16-s + (0.0359 − 0.0494i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (0.888 − 0.288i)3-s + (−0.154 + 0.475i)4-s + (0.768 − 0.639i)5-s + (0.534 + 0.388i)6-s + (−1.80 − 0.585i)7-s + (−0.336 + 0.109i)8-s + (−0.103 + 0.0748i)9-s + (0.685 + 0.173i)10-s + (−0.292 + 0.956i)11-s + 0.467i·12-s + (0.380 + 0.523i)13-s + (−0.414 − 1.27i)14-s + (0.498 − 0.790i)15-s + (−0.202 − 0.146i)16-s + (0.00871 − 0.0119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.930 - 0.365i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.930 - 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44402 + 0.273080i\)
\(L(\frac12)\) \(\approx\) \(1.44402 + 0.273080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + (-1.71 + 1.43i)T \)
11 \( 1 + (0.969 - 3.17i)T \)
good3 \( 1 + (-1.53 + 0.5i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (4.76 + 1.54i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.37 - 1.88i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.0359 + 0.0494i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.636 - 1.96i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.91iT - 23T^{2} \)
29 \( 1 + (-1.27 + 3.93i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.18 + 5.21i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-4.33 - 1.40i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.41 + 4.35i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.61iT - 43T^{2} \)
47 \( 1 + (8.44 - 2.74i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.30 - 1.79i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.599 - 1.84i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.84 - 5.69i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.49iT - 67T^{2} \)
71 \( 1 + (7.13 + 5.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (5.13 + 1.66i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.38 - 3.91i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.49 - 8.94i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.09T + 89T^{2} \)
97 \( 1 + (-4.29 - 5.90i)T + (-29.9 + 92.2i)T^{2} \)
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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

−13.47413137970881978961990667887, −13.23051914873784113710379630671, −12.21010605956935621382902244688, −10.11014906700342421441015239199, −9.388094227372336004849394186541, −8.248817819350152211512542218088, −6.95213298098027797030057950073, −5.96483418296615316158979293021, −4.23195464467006153973302542007, −2.64544296379989623284989780817, 2.93264616595314376130257446237, 3.24236465973866778845405337801, 5.68082789219379366641183597213, 6.55125199528696142651335726961, 8.589362388555827691399768623958, 9.538667972419798967716271173171, 10.21545367169706426341027035292, 11.51363304590525364556986324189, 12.97030590183113386679327673224, 13.47064383439591273792950929702

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