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Properties

Label 2-154-11.3-c1-0-2
Degree $2$
Conductor $154$
Sign $0.834 + 0.551i$
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 + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.224i)5-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (1.61 − 1.17i)9-s − 0.381·10-s + (−1.23 − 3.07i)11-s − 12-s + (4.42 − 3.21i)13-s + (0.309 + 0.951i)14-s + (0.118 − 0.363i)15-s + (−0.809 − 0.587i)16-s + (1.19 + 0.865i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.138 + 0.100i)5-s + (0.330 + 0.239i)6-s + (0.116 − 0.359i)7-s + (0.109 + 0.336i)8-s + (0.539 − 0.391i)9-s − 0.120·10-s + (−0.372 − 0.927i)11-s − 0.288·12-s + (1.22 − 0.892i)13-s + (0.0825 + 0.254i)14-s + (0.0304 − 0.0937i)15-s + (−0.202 − 0.146i)16-s + (0.288 + 0.209i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.834 + 0.551i$
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: \((2,\ 154,\ (\ :1/2),\ 0.834 + 0.551i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835138 - 0.251194i\)
\(L(\frac12)\) \(\approx\) \(0.835138 - 0.251194i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.309 - 0.224i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.42 + 3.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.19 - 0.865i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.07 - 6.37i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 + (2.92 - 9.00i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.42 + 1.76i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.354 - 1.08i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2 - 6.15i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.61T + 43T^{2} \)
47 \( 1 + (-0.5 - 1.53i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.572 - 0.416i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.64 - 5.06i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-11.2 - 8.14i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.70T + 67T^{2} \)
71 \( 1 + (-2.42 - 1.76i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.0450 - 0.138i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.2 + 8.19i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.54 + 5.48i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.09T + 89T^{2} \)
97 \( 1 + (10.0 - 7.33i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.97289018689539537301777541994, −11.83142515411536570340335729267, −10.64645745254764590157496267697, −9.937179977936513656768504493140, −8.415135704924004720326456853831, −7.77217798569400528330282176651, −6.43419526964371737174041694459, −5.66137143729741378826111499866, −3.60494597035643141818533105817, −1.24297128533067861313693951110, 2.00811493960500119815554410277, 3.94939787857411100004136176956, 5.19111003055051646392171911338, 6.82497153392057086101184389962, 8.008711506650867804941462780940, 9.284839131023056107774592389512, 9.884822783089153665282500664318, 11.04903176011263522734624246151, 11.73787407145166809553976282431, 13.01476026079938666903077621864

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