Properties

Label 2-354-177.5-c2-0-5
Degree $2$
Conductor $354$
Sign $-0.999 - 0.0231i$
Analytic cond. $9.64580$
Root an. cond. $3.10576$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.451 + 1.34i)2-s + (−2.93 + 0.624i)3-s + (−1.59 − 1.21i)4-s + (1.01 + 0.404i)5-s + (0.488 − 4.21i)6-s + (0.548 − 0.253i)7-s + (2.34 − 1.58i)8-s + (8.22 − 3.66i)9-s + (−1.00 + 1.17i)10-s + (9.41 + 2.61i)11-s + (5.42 + 2.55i)12-s + (−9.12 + 17.2i)13-s + (0.0924 + 0.849i)14-s + (−3.23 − 0.553i)15-s + (1.07 + 3.85i)16-s + (−3.82 + 8.26i)17-s + ⋯
L(s)  = 1  + (−0.225 + 0.670i)2-s + (−0.978 + 0.208i)3-s + (−0.398 − 0.302i)4-s + (0.203 + 0.0809i)5-s + (0.0813 − 0.702i)6-s + (0.0783 − 0.0362i)7-s + (0.292 − 0.198i)8-s + (0.913 − 0.407i)9-s + (−0.100 + 0.117i)10-s + (0.855 + 0.237i)11-s + (0.452 + 0.213i)12-s + (−0.701 + 1.32i)13-s + (0.00660 + 0.0606i)14-s + (−0.215 − 0.0369i)15-s + (0.0668 + 0.240i)16-s + (−0.225 + 0.486i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0231i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.999 - 0.0231i$
Analytic conductor: \(9.64580\)
Root analytic conductor: \(3.10576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1),\ -0.999 - 0.0231i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00658185 + 0.567570i\)
\(L(\frac12)\) \(\approx\) \(0.00658185 + 0.567570i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.451 - 1.34i)T \)
3 \( 1 + (2.93 - 0.624i)T \)
59 \( 1 + (-11.0 - 57.9i)T \)
good5 \( 1 + (-1.01 - 0.404i)T + (18.1 + 17.1i)T^{2} \)
7 \( 1 + (-0.548 + 0.253i)T + (31.7 - 37.3i)T^{2} \)
11 \( 1 + (-9.41 - 2.61i)T + (103. + 62.3i)T^{2} \)
13 \( 1 + (9.12 - 17.2i)T + (-94.8 - 139. i)T^{2} \)
17 \( 1 + (3.82 - 8.26i)T + (-187. - 220. i)T^{2} \)
19 \( 1 + (-0.315 + 0.0693i)T + (327. - 151. i)T^{2} \)
23 \( 1 + (3.16 + 0.519i)T + (501. + 168. i)T^{2} \)
29 \( 1 + (3.01 + 8.95i)T + (-669. + 508. i)T^{2} \)
31 \( 1 + (16.6 + 3.66i)T + (872. + 403. i)T^{2} \)
37 \( 1 + (25.4 - 37.4i)T + (-506. - 1.27e3i)T^{2} \)
41 \( 1 + (-2.53 + 0.415i)T + (1.59e3 - 536. i)T^{2} \)
43 \( 1 + (-1.54 - 5.57i)T + (-1.58e3 + 953. i)T^{2} \)
47 \( 1 + (28.2 - 11.2i)T + (1.60e3 - 1.51e3i)T^{2} \)
53 \( 1 + (43.1 - 36.6i)T + (454. - 2.77e3i)T^{2} \)
61 \( 1 + (87.5 + 29.4i)T + (2.96e3 + 2.25e3i)T^{2} \)
67 \( 1 + (-7.50 - 11.0i)T + (-1.66e3 + 4.17e3i)T^{2} \)
71 \( 1 + (-51.3 + 20.4i)T + (3.65e3 - 3.46e3i)T^{2} \)
73 \( 1 + (57.2 - 6.22i)T + (5.20e3 - 1.14e3i)T^{2} \)
79 \( 1 + (-34.1 + 20.5i)T + (2.92e3 - 5.51e3i)T^{2} \)
83 \( 1 + (-113. + 6.14i)T + (6.84e3 - 744. i)T^{2} \)
89 \( 1 + (-44.1 - 130. i)T + (-6.30e3 + 4.79e3i)T^{2} \)
97 \( 1 + (88.5 + 9.62i)T + (9.18e3 + 2.02e3i)T^{2} \)
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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

−11.72829494090966587539755857707, −10.73301013302331711387998900641, −9.725348848865572518471527516794, −9.152388730019923819970820319891, −7.74313194677370478719423559540, −6.67876977800115410498900719547, −6.17375760417906960117389494767, −4.86389161557337050772371628295, −4.06652842344908182177969283410, −1.64970919140898370256217280752, 0.32279040189197530520535530537, 1.80085499530220500186775404293, 3.48120361514817860563120777932, 4.88303302808349177747872236515, 5.73263459377932072992094479967, 6.98050233107532541421768455936, 7.940607850462972614674294278738, 9.234182199538365798249257296364, 10.02986949695107764653213366965, 10.91560130618299606366860074557

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