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Properties

Label 2-354-177.176-c5-0-69
Degree $2$
Conductor $354$
Sign $-0.241 + 0.970i$
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 + (−15.5 − 0.434i)3-s + 16·4-s − 38.7i·5-s + (−62.3 − 1.73i)6-s − 41.9·7-s + 64·8-s + (242. + 13.5i)9-s − 154. i·10-s + 166.·11-s + (−249. − 6.95i)12-s + 75.0i·13-s − 167.·14-s + (−16.8 + 603. i)15-s + 256·16-s − 428. i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.999 − 0.0278i)3-s + 0.5·4-s − 0.693i·5-s + (−0.706 − 0.0197i)6-s − 0.323·7-s + 0.353·8-s + (0.998 + 0.0557i)9-s − 0.490i·10-s + 0.414·11-s + (−0.499 − 0.0139i)12-s + 0.123i·13-s − 0.228·14-s + (−0.0193 + 0.692i)15-s + 0.250·16-s − 0.359i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.241 + 0.970i$
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: \((2,\ 354,\ (\ :5/2),\ -0.241 + 0.970i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.813288050\)
\(L(\frac12)\) \(\approx\) \(1.813288050\)
\(L(\frac{7}{2})\) 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 \( 1 + (15.5 + 0.434i)T \)
59 \( 1 + (-5.72e3 + 2.61e4i)T \)
good5 \( 1 + 38.7iT - 3.12e3T^{2} \)
7 \( 1 + 41.9T + 1.68e4T^{2} \)
11 \( 1 - 166.T + 1.61e5T^{2} \)
13 \( 1 - 75.0iT - 3.71e5T^{2} \)
17 \( 1 + 428. iT - 1.41e6T^{2} \)
19 \( 1 + 703.T + 2.47e6T^{2} \)
23 \( 1 - 2.79e3T + 6.43e6T^{2} \)
29 \( 1 + 647. iT - 2.05e7T^{2} \)
31 \( 1 - 3.88e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.00e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.89e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.33e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.91e4T + 2.29e8T^{2} \)
53 \( 1 + 2.59e4iT - 4.18e8T^{2} \)
61 \( 1 + 1.00e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.12e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.99e3iT - 1.80e9T^{2} \)
73 \( 1 + 4.91e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.88e4T + 3.07e9T^{2} \)
83 \( 1 - 4.60e4T + 3.93e9T^{2} \)
89 \( 1 - 8.52e4T + 5.58e9T^{2} \)
97 \( 1 + 1.16e5iT - 8.58e9T^{2} \)
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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

−10.62727394619614208950884176180, −9.607208212041977400102343316793, −8.519594780474267541226525650125, −7.07302467993120163197940316109, −6.45181886949250714020974030340, −5.25929528033298993889892360793, −4.67598252607265922572298933563, −3.42521640810739405922807815594, −1.68409063707694009627650104736, −0.44695420589309966484597157067, 1.21515843669824575641835821357, 2.80471170806495538292071868295, 4.01297542452567260964243813256, 5.03847980088801111293201424831, 6.19261476158472565993045913021, 6.68490063479991593188569235455, 7.71846938017972242502443444801, 9.295757123721815100989888559060, 10.35019432855940676889747493077, 11.05670791549611034379490767179

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