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Properties

Label 2-154-11.3-c1-0-4
Degree $2$
Conductor $154$
Sign $-0.295 + 0.955i$
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 + (−1.02 − 3.16i)3-s + (0.309 − 0.951i)4-s + (2.47 + 1.79i)5-s + (−2.69 − 1.95i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (−6.55 + 4.76i)9-s + 3.05·10-s + (−2.60 + 2.04i)11-s − 3.33·12-s + (1.62 − 1.18i)13-s + (−0.309 − 0.951i)14-s + (3.15 − 9.69i)15-s + (−0.809 − 0.587i)16-s + (3.13 + 2.27i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.594 − 1.82i)3-s + (0.154 − 0.475i)4-s + (1.10 + 0.804i)5-s + (−1.10 − 0.799i)6-s + (0.116 − 0.359i)7-s + (−0.109 − 0.336i)8-s + (−2.18 + 1.58i)9-s + 0.967·10-s + (−0.786 + 0.617i)11-s − 0.962·12-s + (0.451 − 0.327i)13-s + (−0.0825 − 0.254i)14-s + (0.813 − 2.50i)15-s + (−0.202 − 0.146i)16-s + (0.759 + 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.295 + 0.955i$
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: \((2,\ 154,\ (\ :1/2),\ -0.295 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.830879 - 1.12717i\)
\(L(\frac12)\) \(\approx\) \(0.830879 - 1.12717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 - 2.04i)T \)
good3 \( 1 + (1.02 + 3.16i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.47 - 1.79i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-1.62 + 1.18i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.13 - 2.27i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.0995 - 0.306i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.774T + 23T^{2} \)
29 \( 1 + (-0.0934 + 0.287i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.86 + 2.80i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.66 - 5.10i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.88 + 5.80i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 8.28T + 43T^{2} \)
47 \( 1 + (-0.376 - 1.15i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.52 + 1.83i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.16 - 12.8i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (9.57 + 6.95i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.22T + 67T^{2} \)
71 \( 1 + (-11.0 - 7.99i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.70 + 5.26i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (14.2 - 10.3i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.249 - 0.181i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 4.69T + 89T^{2} \)
97 \( 1 + (8.33 - 6.05i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.79615909116695951001466586517, −11.91975487317799286166665127042, −10.81812422809816805387863645825, −10.11611650126991381219765793249, −8.111630895608175735822580981547, −7.03912183758740366310396806630, −6.18104292330237864838874865206, −5.33968262747354690819818429042, −2.77566173915414623639593032887, −1.59886887730278695533608751329, 3.20380010543030768510856674837, 4.75718926117945437472167195868, 5.38024899945685105466735002249, 6.15605438146593236722559830300, 8.441017081293933950643630555178, 9.292039865957692367531135224173, 10.17046233346085740534347091439, 11.19499407681121055260965198688, 12.20126771365872356855050752107, 13.47966747400538382685553367262

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