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Properties

Label 2-354-59.20-c1-0-3
Degree $2$
Conductor $354$
Sign $0.184 - 0.982i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
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  + (0.907 + 0.419i)2-s + (0.947 − 0.319i)3-s + (0.647 + 0.762i)4-s + (−2.35 + 1.41i)5-s + (0.994 + 0.108i)6-s + (0.260 + 4.80i)7-s + (0.267 + 0.963i)8-s + (0.796 − 0.605i)9-s + (−2.73 + 0.297i)10-s + (−0.0646 − 0.394i)11-s + (0.856 + 0.515i)12-s + (−0.265 − 0.201i)13-s + (−1.77 + 4.46i)14-s + (−1.78 + 2.09i)15-s + (−0.161 + 0.986i)16-s + (0.335 − 6.18i)17-s + ⋯
L(s)  = 1  + (0.641 + 0.296i)2-s + (0.547 − 0.184i)3-s + (0.323 + 0.381i)4-s + (−1.05 + 0.634i)5-s + (0.405 + 0.0441i)6-s + (0.0983 + 1.81i)7-s + (0.0945 + 0.340i)8-s + (0.265 − 0.201i)9-s + (−0.864 + 0.0940i)10-s + (−0.0194 − 0.118i)11-s + (0.247 + 0.148i)12-s + (−0.0736 − 0.0559i)13-s + (−0.475 + 1.19i)14-s + (−0.459 + 0.541i)15-s + (−0.0404 + 0.246i)16-s + (0.0812 − 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.184 - 0.982i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.184 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47818 + 1.22650i\)
\(L(\frac12)\) \(\approx\) \(1.47818 + 1.22650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.907 - 0.419i)T \)
3 \( 1 + (-0.947 + 0.319i)T \)
59 \( 1 + (5.19 - 5.65i)T \)
good5 \( 1 + (2.35 - 1.41i)T + (2.34 - 4.41i)T^{2} \)
7 \( 1 + (-0.260 - 4.80i)T + (-6.95 + 0.756i)T^{2} \)
11 \( 1 + (0.0646 + 0.394i)T + (-10.4 + 3.51i)T^{2} \)
13 \( 1 + (0.265 + 0.201i)T + (3.47 + 12.5i)T^{2} \)
17 \( 1 + (-0.335 + 6.18i)T + (-16.9 - 1.83i)T^{2} \)
19 \( 1 + (-2.11 - 2.00i)T + (1.02 + 18.9i)T^{2} \)
23 \( 1 + (-1.98 - 0.437i)T + (20.8 + 9.65i)T^{2} \)
29 \( 1 + (-7.13 + 3.29i)T + (18.7 - 22.1i)T^{2} \)
31 \( 1 + (-1.64 + 1.56i)T + (1.67 - 30.9i)T^{2} \)
37 \( 1 + (1.48 - 5.34i)T + (-31.7 - 19.0i)T^{2} \)
41 \( 1 + (-7.16 + 1.57i)T + (37.2 - 17.2i)T^{2} \)
43 \( 1 + (-1.70 + 10.4i)T + (-40.7 - 13.7i)T^{2} \)
47 \( 1 + (2.53 + 1.52i)T + (22.0 + 41.5i)T^{2} \)
53 \( 1 + (5.20 + 0.565i)T + (51.7 + 11.3i)T^{2} \)
61 \( 1 + (6.26 + 2.90i)T + (39.4 + 46.4i)T^{2} \)
67 \( 1 + (-3.04 - 10.9i)T + (-57.4 + 34.5i)T^{2} \)
71 \( 1 + (-1.34 - 0.810i)T + (33.2 + 62.7i)T^{2} \)
73 \( 1 + (-3.49 + 8.77i)T + (-52.9 - 50.2i)T^{2} \)
79 \( 1 + (-10.7 - 3.60i)T + (62.8 + 47.8i)T^{2} \)
83 \( 1 + (7.76 + 11.4i)T + (-30.7 + 77.1i)T^{2} \)
89 \( 1 + (10.8 - 5.02i)T + (57.6 - 67.8i)T^{2} \)
97 \( 1 + (-1.82 - 4.56i)T + (-70.4 + 66.7i)T^{2} \)
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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

−11.99885519045516182461652010602, −11.15180530624812590600986379630, −9.649300078650374912121706306697, −8.652018897140355169009826382903, −7.84937503892500822751024877932, −6.96643571517882416603956538749, −5.82072556805309254886286585669, −4.72747330066285305743666908901, −3.26012138923111790595743219888, −2.55257899262566018741189111307, 1.15259267897184515087792316902, 3.27962353841894968944601469810, 4.19337135973977192197779534554, 4.73721934094526184389248978976, 6.55643201499963694430161313147, 7.60056312338756515200424216092, 8.237848130759296551928679986618, 9.558107051993814159550028622752, 10.62063605526516758035634594779, 11.12539210586688365069503136872

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