Properties

Label 2-354-177.176-c5-0-85
Degree $2$
Conductor $354$
Sign $-0.611 - 0.791i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + (−11.6 − 10.3i)3-s + 16·4-s − 106. i·5-s + (46.7 + 41.2i)6-s + 71.9·7-s − 64·8-s + (30.3 + 241. i)9-s + 424. i·10-s + 484.·11-s + (−187. − 164. i)12-s + 699. i·13-s − 287.·14-s + (−1.09e3 + 1.24e3i)15-s + 256·16-s − 2.02e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.749 − 0.661i)3-s + 0.5·4-s − 1.89i·5-s + (0.530 + 0.467i)6-s + 0.554·7-s − 0.353·8-s + (0.124 + 0.992i)9-s + 1.34i·10-s + 1.20·11-s + (−0.374 − 0.330i)12-s + 1.14i·13-s − 0.392·14-s + (−1.25 + 1.42i)15-s + 0.250·16-s − 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.611 - 0.791i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ -0.611 - 0.791i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4956147964\)
\(L(\frac12)\) \(\approx\) \(0.4956147964\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
3 \( 1 + (11.6 + 10.3i)T \)
59 \( 1 + (2.62e4 + 5.05e3i)T \)
good5 \( 1 + 106. iT - 3.12e3T^{2} \)
7 \( 1 - 71.9T + 1.68e4T^{2} \)
11 \( 1 - 484.T + 1.61e5T^{2} \)
13 \( 1 - 699. iT - 3.71e5T^{2} \)
17 \( 1 + 2.02e3iT - 1.41e6T^{2} \)
19 \( 1 - 219.T + 2.47e6T^{2} \)
23 \( 1 + 2.06e3T + 6.43e6T^{2} \)
29 \( 1 - 259. iT - 2.05e7T^{2} \)
31 \( 1 + 3.94e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.22e4iT - 6.93e7T^{2} \)
41 \( 1 + 6.33e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.69e4iT - 1.47e8T^{2} \)
47 \( 1 + 7.33e3T + 2.29e8T^{2} \)
53 \( 1 - 2.84e4iT - 4.18e8T^{2} \)
61 \( 1 + 5.27e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.08e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.27e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.01e4iT - 2.07e9T^{2} \)
79 \( 1 + 8.88e4T + 3.07e9T^{2} \)
83 \( 1 + 8.31e4T + 3.93e9T^{2} \)
89 \( 1 - 4.70e4T + 5.58e9T^{2} \)
97 \( 1 - 1.47e5iT - 8.58e9T^{2} \)
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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

−9.734539399623068914382900632610, −9.098560671386075974790354766019, −8.254921405382007422075904882639, −7.30604417448970167705372879254, −6.24279906192468441164515951857, −5.10713908483718709321357105196, −4.33557821048500786367600536993, −1.87303910393844855267274037050, −1.18532671006261776944496831942, −0.19403974572259381307774953614, 1.55595926043212942220484364431, 3.13374166928655333449782915326, 4.03854111638217484659920261346, 5.85613034905976950085313337195, 6.38294150214518242658773097158, 7.39799583801156260448858580955, 8.448104411491625789423301383375, 9.822491987861063069606263063693, 10.33638491159057227235232076528, 11.07542260935247976485879850828

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