Properties

Label 2-354-59.21-c1-0-7
Degree $2$
Conductor $354$
Sign $0.966 - 0.257i$
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  + (0.856 + 0.515i)2-s + (0.370 − 0.928i)3-s + (0.468 + 0.883i)4-s + (−1.41 − 0.153i)5-s + (0.796 − 0.605i)6-s + (2.98 + 1.00i)7-s + (−0.0541 + 0.998i)8-s + (−0.725 − 0.687i)9-s + (−1.13 − 0.860i)10-s + (2.18 + 3.23i)11-s + (0.994 − 0.108i)12-s + (3.44 − 3.26i)13-s + (2.03 + 2.39i)14-s + (−0.666 + 1.25i)15-s + (−0.561 + 0.827i)16-s + (1.15 − 0.388i)17-s + ⋯
L(s)  = 1  + (0.605 + 0.364i)2-s + (0.213 − 0.536i)3-s + (0.234 + 0.441i)4-s + (−0.632 − 0.0687i)5-s + (0.325 − 0.247i)6-s + (1.12 + 0.379i)7-s + (−0.0191 + 0.353i)8-s + (−0.241 − 0.229i)9-s + (−0.358 − 0.272i)10-s + (0.660 + 0.973i)11-s + (0.286 − 0.0312i)12-s + (0.956 − 0.906i)13-s + (0.544 + 0.640i)14-s + (−0.172 + 0.324i)15-s + (−0.140 + 0.206i)16-s + (0.279 − 0.0942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.966 - 0.257i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02666 + 0.264900i\)
\(L(\frac12)\) \(\approx\) \(2.02666 + 0.264900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.856 - 0.515i)T \)
3 \( 1 + (-0.370 + 0.928i)T \)
59 \( 1 + (7.47 + 1.74i)T \)
good5 \( 1 + (1.41 + 0.153i)T + (4.88 + 1.07i)T^{2} \)
7 \( 1 + (-2.98 - 1.00i)T + (5.57 + 4.23i)T^{2} \)
11 \( 1 + (-2.18 - 3.23i)T + (-4.07 + 10.2i)T^{2} \)
13 \( 1 + (-3.44 + 3.26i)T + (0.703 - 12.9i)T^{2} \)
17 \( 1 + (-1.15 + 0.388i)T + (13.5 - 10.2i)T^{2} \)
19 \( 1 + (-0.484 + 2.95i)T + (-18.0 - 6.06i)T^{2} \)
23 \( 1 + (0.198 - 0.715i)T + (-19.7 - 11.8i)T^{2} \)
29 \( 1 + (5.78 - 3.47i)T + (13.5 - 25.6i)T^{2} \)
31 \( 1 + (0.539 + 3.28i)T + (-29.3 + 9.89i)T^{2} \)
37 \( 1 + (-0.202 - 3.73i)T + (-36.7 + 4.00i)T^{2} \)
41 \( 1 + (1.29 + 4.65i)T + (-35.1 + 21.1i)T^{2} \)
43 \( 1 + (4.27 - 6.31i)T + (-15.9 - 39.9i)T^{2} \)
47 \( 1 + (8.23 - 0.895i)T + (45.9 - 10.1i)T^{2} \)
53 \( 1 + (2.61 - 1.99i)T + (14.1 - 51.0i)T^{2} \)
61 \( 1 + (-3.52 - 2.11i)T + (28.5 + 53.8i)T^{2} \)
67 \( 1 + (-0.226 + 4.17i)T + (-66.6 - 7.24i)T^{2} \)
71 \( 1 + (-13.5 + 1.47i)T + (69.3 - 15.2i)T^{2} \)
73 \( 1 + (-3.86 - 4.54i)T + (-11.8 + 72.0i)T^{2} \)
79 \( 1 + (5.26 + 13.2i)T + (-57.3 + 54.3i)T^{2} \)
83 \( 1 + (-3.07 - 1.42i)T + (53.7 + 63.2i)T^{2} \)
89 \( 1 + (0.392 - 0.236i)T + (41.6 - 78.6i)T^{2} \)
97 \( 1 + (4.23 - 4.98i)T + (-15.6 - 95.7i)T^{2} \)
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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

−11.63297757405005720625419884432, −11.04008699302717686198195031423, −9.463334572116202776980087256255, −8.290561126004005395180512459523, −7.79007492972866882265935537763, −6.78926846207104079335153239091, −5.59448273239318538481330295252, −4.56165981306967855374747201893, −3.37946449678972033200825863853, −1.72869848672394035347323587512, 1.61579712809850948848093730659, 3.59010450752569302078601873551, 4.06714354806145797717598957522, 5.30445943000924868338275707029, 6.44634714175897198707934025514, 7.82624630725474270168508986766, 8.592241395247414160250191085093, 9.696945080427124496677351502302, 10.97319025970856633365920212170, 11.29329001797178422163397509933

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