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Properties

Label 2-154-11.4-c1-0-3
Degree $2$
Conductor $154$
Sign $0.939 - 0.342i$
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 + (0.220 − 0.679i)3-s + (0.309 + 0.951i)4-s + (0.451 − 0.328i)5-s + (0.578 − 0.420i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (2.01 + 1.46i)9-s + 0.558·10-s + (1.91 − 2.70i)11-s + 0.714·12-s + (−3.67 − 2.66i)13-s + (−0.309 + 0.951i)14-s + (−0.123 − 0.379i)15-s + (−0.809 + 0.587i)16-s + (−5.44 + 3.95i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.127 − 0.392i)3-s + (0.154 + 0.475i)4-s + (0.201 − 0.146i)5-s + (0.236 − 0.171i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (0.671 + 0.487i)9-s + 0.176·10-s + (0.578 − 0.816i)11-s + 0.206·12-s + (−1.01 − 0.739i)13-s + (−0.0825 + 0.254i)14-s + (−0.0318 − 0.0979i)15-s + (−0.202 + 0.146i)16-s + (−1.31 + 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.939 - 0.342i$
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: \((2,\ 154,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56450 + 0.276568i\)
\(L(\frac12)\) \(\approx\) \(1.56450 + 0.276568i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 + 2.70i)T \)
good3 \( 1 + (-0.220 + 0.679i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.451 + 0.328i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (3.67 + 2.66i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.44 - 3.95i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.872 + 2.68i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.77T + 23T^{2} \)
29 \( 1 + (1.93 + 5.94i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.43 + 1.04i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.16 - 9.72i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.38 + 4.26i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.188T + 43T^{2} \)
47 \( 1 + (-3.65 + 11.2i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.27 - 0.929i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.85 - 8.79i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.25 - 1.63i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.36T + 67T^{2} \)
71 \( 1 + (-9.75 + 7.08i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.68 - 14.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.68 - 1.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.75 + 1.99i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 8.40T + 89T^{2} \)
97 \( 1 + (-6.78 - 4.93i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.25267259528374118553387870615, −12.25521147943204526084828440483, −11.25377845923821954764325856655, −9.957588417296996845482076399940, −8.638218513170863683345116641956, −7.68491316906794227010678687812, −6.53523619527744305831396549762, −5.41751781824761196291114355034, −4.09318858661731543724910863366, −2.26401140672972347514027091273, 2.10880567635175124580303990422, 3.97800001387445315890539175838, 4.74198108150658984673783147816, 6.47070288606721861041739660733, 7.37374842870240854055276490548, 9.311427446166956958159271115834, 9.799973677063411505450793974731, 10.95203889484263482502222892567, 12.06317550548895216589890602456, 12.74062559290582453748784168467

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