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Properties

Label 2-354-177.176-c3-0-2
Degree $2$
Conductor $354$
Sign $-0.540 - 0.841i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + (1.05 − 5.08i)3-s + 4·4-s + 6.49i·5-s + (2.10 − 10.1i)6-s − 27.4·7-s + 8·8-s + (−24.7 − 10.6i)9-s + 12.9i·10-s − 15.6·11-s + (4.20 − 20.3i)12-s + 50.8i·13-s − 54.9·14-s + (33.0 + 6.82i)15-s + 16·16-s + 87.9i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.202 − 0.979i)3-s + 0.5·4-s + 0.580i·5-s + (0.143 − 0.692i)6-s − 1.48·7-s + 0.353·8-s + (−0.918 − 0.396i)9-s + 0.410i·10-s − 0.430·11-s + (0.101 − 0.489i)12-s + 1.08i·13-s − 1.04·14-s + (0.568 + 0.117i)15-s + 0.250·16-s + 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ -0.540 - 0.841i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6708763057\)
\(L(\frac12)\) \(\approx\) \(0.6708763057\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + (-1.05 + 5.08i)T \)
59 \( 1 + (323. - 317. i)T \)
good5 \( 1 - 6.49iT - 125T^{2} \)
7 \( 1 + 27.4T + 343T^{2} \)
11 \( 1 + 15.6T + 1.33e3T^{2} \)
13 \( 1 - 50.8iT - 2.19e3T^{2} \)
17 \( 1 - 87.9iT - 4.91e3T^{2} \)
19 \( 1 + 54.3T + 6.85e3T^{2} \)
23 \( 1 + 138.T + 1.21e4T^{2} \)
29 \( 1 - 24.5iT - 2.43e4T^{2} \)
31 \( 1 + 176. iT - 2.97e4T^{2} \)
37 \( 1 + 128. iT - 5.06e4T^{2} \)
41 \( 1 - 27.5iT - 6.89e4T^{2} \)
43 \( 1 - 121. iT - 7.95e4T^{2} \)
47 \( 1 + 353.T + 1.03e5T^{2} \)
53 \( 1 - 249. iT - 1.48e5T^{2} \)
61 \( 1 + 441. iT - 2.26e5T^{2} \)
67 \( 1 + 148. iT - 3.00e5T^{2} \)
71 \( 1 - 259. iT - 3.57e5T^{2} \)
73 \( 1 - 229. iT - 3.89e5T^{2} \)
79 \( 1 + 4.13T + 4.93e5T^{2} \)
83 \( 1 + 1.17e3T + 5.71e5T^{2} \)
89 \( 1 - 305.T + 7.04e5T^{2} \)
97 \( 1 + 1.12e3iT - 9.12e5T^{2} \)
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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

−11.57513480304506104480878794545, −10.59437879373995112128596492013, −9.567329334948022773081378935945, −8.373181377097695939059134922519, −7.25122291363943320837355062030, −6.39657278372319689408585868918, −5.99551249968589188309991357525, −4.08038768074171158724044591702, −3.03070195669944902690628854217, −1.96649922750297014069086424440, 0.16178811335001583386830835866, 2.72140706298814011971150464461, 3.49592178685444498711193614041, 4.72999006534110253475295989682, 5.56144658939871629179364195927, 6.60793726395091658136761773246, 7.986179537722385031687272211143, 9.024087172726213737206654625842, 9.977789788066581494868287994447, 10.51589212476803215903738719092

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