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Properties

Label 2-154-11.3-c1-0-1
Degree $2$
Conductor $154$
Sign $0.927 - 0.374i$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
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  + (0.809 − 0.587i)2-s + (0.690 + 2.12i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.224i)5-s + (1.80 + 1.31i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−1.61 + 1.17i)9-s + 0.381·10-s + (−3.23 − 0.726i)11-s + 2.23·12-s + (2.42 − 1.76i)13-s + (0.309 + 0.951i)14-s + (−0.263 + 0.812i)15-s + (−0.809 − 0.587i)16-s + (2.80 + 2.04i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.398 + 1.22i)3-s + (0.154 − 0.475i)4-s + (0.138 + 0.100i)5-s + (0.738 + 0.536i)6-s + (−0.116 + 0.359i)7-s + (−0.109 − 0.336i)8-s + (−0.539 + 0.391i)9-s + 0.120·10-s + (−0.975 − 0.219i)11-s + 0.645·12-s + (0.673 − 0.489i)13-s + (0.0825 + 0.254i)14-s + (−0.0681 + 0.209i)15-s + (−0.202 − 0.146i)16-s + (0.681 + 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60395 + 0.311545i\)
\(L(\frac12)\) \(\approx\) \(1.60395 + 0.311545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (3.23 + 0.726i)T \)
good3 \( 1 + (-0.690 - 2.12i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.309 - 0.224i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-2.42 + 1.76i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.80 - 2.04i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.69 + 5.20i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 + (-0.927 + 2.85i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.80 - 2.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.64 - 5.06i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.472 + 1.45i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.85T + 43T^{2} \)
47 \( 1 + (-0.354 - 1.08i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.66 - 4.84i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.5 - 13.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-7.97 - 5.79i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 + (3.42 + 2.48i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.57 + 10.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.89 + 7.19i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.92 - 1.40i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 4.85T + 89T^{2} \)
97 \( 1 + (-6.85 + 4.97i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.11558480706021617184825732909, −12.04284746992987863528405068501, −10.70325197379324840295156126243, −10.28273485347249215508177009752, −9.161483535805927021574403767802, −8.062641475744388296339096018768, −6.17413973142638677137434322744, −5.06901909675096844881221667570, −3.86336671528832794790726729471, −2.70078214842645652357893827405, 1.96747713985001319669055969923, 3.68684481036945238573681167265, 5.42517292164450075958597838659, 6.57144326751455814840562263475, 7.61230626080186889879908835153, 8.229034335282274941454894211379, 9.783267641425000470853213042845, 11.15478227071123040180060611321, 12.45229111734744006511894668263, 12.90571846852843471446031178457

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