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Properties

Label 2-354-177.176-c5-0-46
Degree $2$
Conductor $354$
Sign $0.751 + 0.660i$
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 + (−11.5 − 10.4i)3-s + 16·4-s − 67.9i·5-s + (−46.0 − 41.9i)6-s − 61.6·7-s + 64·8-s + (22.5 + 241. i)9-s − 271. i·10-s + 640.·11-s + (−184. − 167. i)12-s + 578. i·13-s − 246.·14-s + (−713. + 783. i)15-s + 256·16-s + 1.37e3i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.739 − 0.673i)3-s + 0.5·4-s − 1.21i·5-s + (−0.522 − 0.476i)6-s − 0.475·7-s + 0.353·8-s + (0.0928 + 0.995i)9-s − 0.859i·10-s + 1.59·11-s + (−0.369 − 0.336i)12-s + 0.948i·13-s − 0.336·14-s + (−0.819 + 0.899i)15-s + 0.250·16-s + 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.751 + 0.660i$
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: \((2,\ 354,\ (\ :5/2),\ 0.751 + 0.660i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.852030142\)
\(L(\frac12)\) \(\approx\) \(2.852030142\)
\(L(\frac{7}{2})\) 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 \( 1 + (11.5 + 10.4i)T \)
59 \( 1 + (2.67e4 - 479. i)T \)
good5 \( 1 + 67.9iT - 3.12e3T^{2} \)
7 \( 1 + 61.6T + 1.68e4T^{2} \)
11 \( 1 - 640.T + 1.61e5T^{2} \)
13 \( 1 - 578. iT - 3.71e5T^{2} \)
17 \( 1 - 1.37e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.52e3T + 2.47e6T^{2} \)
23 \( 1 - 625.T + 6.43e6T^{2} \)
29 \( 1 - 5.93e3iT - 2.05e7T^{2} \)
31 \( 1 - 491. iT - 2.86e7T^{2} \)
37 \( 1 + 1.35e4iT - 6.93e7T^{2} \)
41 \( 1 - 780. iT - 1.15e8T^{2} \)
43 \( 1 - 4.53e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.83e4T + 2.29e8T^{2} \)
53 \( 1 + 1.72e4iT - 4.18e8T^{2} \)
61 \( 1 + 854. iT - 8.44e8T^{2} \)
67 \( 1 - 2.85e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.29e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.07e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.94e4T + 3.07e9T^{2} \)
83 \( 1 + 6.49e4T + 3.93e9T^{2} \)
89 \( 1 - 1.22e5T + 5.58e9T^{2} \)
97 \( 1 - 828. iT - 8.58e9T^{2} \)
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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

−10.91604578656542590491137570298, −9.508038664014965540412066247242, −8.727980398002218570876020955073, −7.36447059395694396659701114131, −6.52778779013843312819495239784, −5.65756355789908286478937776918, −4.68360996963664062594395695688, −3.66070515936897012940371466651, −1.71249205936259147834240778328, −0.976427951987151853975549330436, 0.852388977364535115054173839409, 2.95128844902441721243656247595, 3.56990178629592389261791003252, 4.81864152385804198140459607682, 5.95604144123867885097645225522, 6.62768740995401421967327501098, 7.48210875765575559164023991114, 9.330410816132863719462189604812, 9.958904474977909341763060959881, 10.92079442243139400301727425153

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