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Properties

Label 2-354-177.176-c1-0-8
Degree $2$
Conductor $354$
Sign $0.779 - 0.625i$
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 + (−1.43 + 0.975i)3-s + 4-s − 0.0999i·5-s + (−1.43 + 0.975i)6-s + 2.48·7-s + 8-s + (1.09 − 2.79i)9-s − 0.0999i·10-s + 1.50·11-s + (−1.43 + 0.975i)12-s + 3.74i·13-s + 2.48·14-s + (0.0974 + 0.143i)15-s + 16-s + 6.68i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.826 + 0.563i)3-s + 0.5·4-s − 0.0446i·5-s + (−0.584 + 0.398i)6-s + 0.939·7-s + 0.353·8-s + (0.365 − 0.930i)9-s − 0.0315i·10-s + 0.453·11-s + (−0.413 + 0.281i)12-s + 1.03i·13-s + 0.664·14-s + (0.0251 + 0.0369i)15-s + 0.250·16-s + 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.779 - 0.625i$
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: \((2,\ 354,\ (\ :1/2),\ 0.779 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65275 + 0.581052i\)
\(L(\frac12)\) \(\approx\) \(1.65275 + 0.581052i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 - 0.975i)T \)
59 \( 1 + (7.65 - 0.598i)T \)
good5 \( 1 + 0.0999iT - 5T^{2} \)
7 \( 1 - 2.48T + 7T^{2} \)
11 \( 1 - 1.50T + 11T^{2} \)
13 \( 1 - 3.74iT - 13T^{2} \)
17 \( 1 - 6.68iT - 17T^{2} \)
19 \( 1 - 3.99T + 19T^{2} \)
23 \( 1 + 7.31T + 23T^{2} \)
29 \( 1 + 3.18iT - 29T^{2} \)
31 \( 1 + 9.23iT - 31T^{2} \)
37 \( 1 - 0.665iT - 37T^{2} \)
41 \( 1 + 3.99iT - 41T^{2} \)
43 \( 1 + 10.0iT - 43T^{2} \)
47 \( 1 + 4.30T + 47T^{2} \)
53 \( 1 + 0.250iT - 53T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 - 7.06iT - 67T^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + 9.13T + 79T^{2} \)
83 \( 1 + 5.43T + 83T^{2} \)
89 \( 1 - 8.23T + 89T^{2} \)
97 \( 1 + 3.83iT - 97T^{2} \)
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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

−11.75388938985474895591604878718, −10.85458521682325494638961674892, −10.00790941661205236831365211149, −8.851785940028270551461812080096, −7.63414513468364428641789677212, −6.40406513425582009907312558876, −5.66001137628711966352222729782, −4.48356315150511407749536223809, −3.85920465020403360410838660233, −1.74147727734882790774043167309, 1.33229144942458229243096498656, 3.00259817406281952536928454367, 4.76536806810925632490297498867, 5.28680029448147551597641943007, 6.46418438189130720091080111394, 7.40541734147381021138261515233, 8.207323465900466133112851963916, 9.781691155812193175764790018234, 10.85579134144773964981587907940, 11.52297557268960530476710322392

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