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Properties

Label 2-354-59.19-c1-0-8
Degree $2$
Conductor $354$
Sign $-0.0141 + 0.999i$
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  + (0.468 − 0.883i)2-s + (0.725 − 0.687i)3-s + (−0.561 − 0.827i)4-s + (2.03 − 0.448i)5-s + (−0.267 − 0.963i)6-s + (−0.284 + 0.216i)7-s + (−0.994 + 0.108i)8-s + (0.0541 − 0.998i)9-s + (0.558 − 2.01i)10-s + (−1.78 − 4.48i)11-s + (−0.976 − 0.214i)12-s + (0.0677 + 1.25i)13-s + (0.0578 + 0.353i)14-s + (1.17 − 1.72i)15-s + (−0.370 + 0.928i)16-s + (5.16 + 3.92i)17-s + ⋯
L(s)  = 1  + (0.331 − 0.624i)2-s + (0.419 − 0.397i)3-s + (−0.280 − 0.413i)4-s + (0.911 − 0.200i)5-s + (−0.109 − 0.393i)6-s + (−0.107 + 0.0818i)7-s + (−0.351 + 0.0382i)8-s + (0.0180 − 0.332i)9-s + (0.176 − 0.636i)10-s + (−0.538 − 1.35i)11-s + (−0.281 − 0.0620i)12-s + (0.0187 + 0.346i)13-s + (0.0154 + 0.0943i)14-s + (0.302 − 0.446i)15-s + (−0.0925 + 0.232i)16-s + (1.25 + 0.952i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0141 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0141 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.0141 + 0.999i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ -0.0141 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35250 - 1.37177i\)
\(L(\frac12)\) \(\approx\) \(1.35250 - 1.37177i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.468 + 0.883i)T \)
3 \( 1 + (-0.725 + 0.687i)T \)
59 \( 1 + (-5.90 - 4.90i)T \)
good5 \( 1 + (-2.03 + 0.448i)T + (4.53 - 2.09i)T^{2} \)
7 \( 1 + (0.284 - 0.216i)T + (1.87 - 6.74i)T^{2} \)
11 \( 1 + (1.78 + 4.48i)T + (-7.98 + 7.56i)T^{2} \)
13 \( 1 + (-0.0677 - 1.25i)T + (-12.9 + 1.40i)T^{2} \)
17 \( 1 + (-5.16 - 3.92i)T + (4.54 + 16.3i)T^{2} \)
19 \( 1 + (-2.53 + 0.855i)T + (15.1 - 11.4i)T^{2} \)
23 \( 1 + (2.75 - 1.65i)T + (10.7 - 20.3i)T^{2} \)
29 \( 1 + (1.24 + 2.35i)T + (-16.2 + 24.0i)T^{2} \)
31 \( 1 + (1.75 + 0.592i)T + (24.6 + 18.7i)T^{2} \)
37 \( 1 + (9.72 + 1.05i)T + (36.1 + 7.95i)T^{2} \)
41 \( 1 + (-7.74 - 4.66i)T + (19.2 + 36.2i)T^{2} \)
43 \( 1 + (0.156 - 0.391i)T + (-31.2 - 29.5i)T^{2} \)
47 \( 1 + (-9.69 - 2.13i)T + (42.6 + 19.7i)T^{2} \)
53 \( 1 + (-1.99 - 7.17i)T + (-45.4 + 27.3i)T^{2} \)
61 \( 1 + (-2.50 + 4.72i)T + (-34.2 - 50.4i)T^{2} \)
67 \( 1 + (9.01 - 0.980i)T + (65.4 - 14.4i)T^{2} \)
71 \( 1 + (-1.30 - 0.287i)T + (64.4 + 29.8i)T^{2} \)
73 \( 1 + (-1.58 - 9.67i)T + (-69.1 + 23.3i)T^{2} \)
79 \( 1 + (1.00 + 0.955i)T + (4.27 + 78.8i)T^{2} \)
83 \( 1 + (5.06 - 5.95i)T + (-13.4 - 81.9i)T^{2} \)
89 \( 1 + (0.128 + 0.242i)T + (-49.9 + 73.6i)T^{2} \)
97 \( 1 + (-1.09 + 6.64i)T + (-91.9 - 30.9i)T^{2} \)
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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

−11.27623017385498334184906920560, −10.31233993107880603959466065849, −9.487377033035122927530069105206, −8.602310629152177752936282718097, −7.57245079820871440980530088334, −5.97118649883167559334804372242, −5.58031170917694754345784954089, −3.82849861362792334345029673875, −2.72618195292239254811655854661, −1.36999319861575556411191094611, 2.26473632096682325943752387840, 3.58653271792177303210493839145, 5.00097580524195933970599879661, 5.67086295186392303656179773412, 7.06335886789967154859665183595, 7.72322212375111609376588388409, 8.989877413929704053542997301698, 9.931363261861642589463422803847, 10.31964216212742080608095686521, 11.95226719785886336579446496027

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