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Properties

Label 2-354-177.176-c1-0-13
Degree $2$
Conductor $354$
Sign $0.987 - 0.156i$
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  + 2-s + (1.64 − 0.548i)3-s + 4-s + 2.54i·5-s + (1.64 − 0.548i)6-s − 0.0900·7-s + 8-s + (2.39 − 1.80i)9-s + 2.54i·10-s − 2.40·11-s + (1.64 − 0.548i)12-s − 2.81i·13-s − 0.0900·14-s + (1.39 + 4.18i)15-s + 16-s + 3.17i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.948 − 0.316i)3-s + 0.5·4-s + 1.13i·5-s + (0.670 − 0.223i)6-s − 0.0340·7-s + 0.353·8-s + (0.799 − 0.600i)9-s + 0.805i·10-s − 0.725·11-s + (0.474 − 0.158i)12-s − 0.781i·13-s − 0.0240·14-s + (0.360 + 1.08i)15-s + 0.250·16-s + 0.769i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.987 - 0.156i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ 0.987 - 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.57546 + 0.203005i\)
\(L(\frac12)\) \(\approx\) \(2.57546 + 0.203005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.64 + 0.548i)T \)
59 \( 1 + (-7.57 - 1.26i)T \)
good5 \( 1 - 2.54iT - 5T^{2} \)
7 \( 1 + 0.0900T + 7T^{2} \)
11 \( 1 + 2.40T + 11T^{2} \)
13 \( 1 + 2.81iT - 13T^{2} \)
17 \( 1 - 3.17iT - 17T^{2} \)
19 \( 1 + 2.49T + 19T^{2} \)
23 \( 1 + 3.09T + 23T^{2} \)
29 \( 1 + 6.21iT - 29T^{2} \)
31 \( 1 - 2.16iT - 31T^{2} \)
37 \( 1 + 11.5iT - 37T^{2} \)
41 \( 1 + 0.762iT - 41T^{2} \)
43 \( 1 - 7.62iT - 43T^{2} \)
47 \( 1 + 7.90T + 47T^{2} \)
53 \( 1 + 3.58iT - 53T^{2} \)
61 \( 1 - 3.56iT - 61T^{2} \)
67 \( 1 - 14.5iT - 67T^{2} \)
71 \( 1 - 4.84iT - 71T^{2} \)
73 \( 1 - 6.03iT - 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 - 1.68T + 83T^{2} \)
89 \( 1 - 8.62T + 89T^{2} \)
97 \( 1 - 0.394iT - 97T^{2} \)
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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

−11.52394665060801365777608396483, −10.52939028983189268878100086903, −9.915999715711424544949723454800, −8.395900811690613935017707528294, −7.66006951330901103817309775899, −6.72627157320879673285975909248, −5.77656055483808065702856509678, −4.14741281119110861241262499869, −3.11604809754873204705706354911, −2.21577488532363537192228575653, 1.86838842899287159893391551978, 3.25714664459595542053356796462, 4.54133809198903116450312506711, 5.08970886448939221275757615201, 6.63972916690878810340775919089, 7.82929762556591590154642289006, 8.634894453309296527292330019795, 9.503400684468141405076302126477, 10.45806577870915287190212012621, 11.65419458288311011327686449148

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