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Properties

Label 2-154-11.4-c1-0-4
Degree $2$
Conductor $154$
Sign $-0.220 + 0.975i$
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 + (1.30 − 0.951i)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 − 1.61·10-s + (1.23 + 3.07i)11-s + 2.23·12-s + (−3.42 − 2.48i)13-s + (−0.309 + 0.951i)14-s + (−1.11 − 3.44i)15-s + (−0.809 + 0.587i)16-s + (−3.42 + 2.48i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.398 − 1.22i)3-s + (0.154 + 0.475i)4-s + (0.585 − 0.425i)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 − 0.511·10-s + (0.372 + 0.927i)11-s + 0.645·12-s + (−0.950 − 0.690i)13-s + (−0.0825 + 0.254i)14-s + (−0.288 − 0.888i)15-s + (−0.202 + 0.146i)16-s + (−0.831 + 0.603i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640742 - 0.802017i\)
\(L(\frac12)\) \(\approx\) \(0.640742 - 0.802017i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 3.07i)T \)
good3 \( 1 + (-0.690 + 2.12i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.30 + 0.951i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (3.42 + 2.48i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.42 - 2.48i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.54 + 4.75i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (-2.30 - 7.10i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.80 - 2.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.354 + 1.08i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.76 - 8.50i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.85T + 43T^{2} \)
47 \( 1 + (-1.35 + 4.16i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.04 + 2.21i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.97 - 6.06i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.35 + 6.06i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + (-3.66 + 2.66i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.28 - 7.02i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (12.1 + 8.81i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.16 + 3.75i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + (10.8 + 7.88i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.93182331633967737428652409942, −11.88826693095271021325373453826, −10.55590797361368923413414123275, −9.510684573764461861196136294009, −8.578479193267792296817885688667, −7.35912249925035080040701577095, −6.74501981012713101069932431856, −4.85148347275503476530672597704, −2.71186647221147019690139333442, −1.38439070417398012573350215558, 2.65324428284340386429079928002, 4.31827717781268752027547242588, 5.71256827381772749021015497002, 6.85787582651727606709291841006, 8.385231430761263709868956239975, 9.369467306843761357268226920017, 9.868592757155766299665162641695, 10.89069114462334815938757678260, 11.98030361230194923507316727150, 13.76126423202843157003393782576

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