Properties

Label 2-354-59.58-c2-0-8
Degree $2$
Conductor $354$
Sign $0.975 + 0.221i$
Analytic cond. $9.64580$
Root an. cond. $3.10576$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.73·3-s − 2.00·4-s + 6.82·5-s + 2.44i·6-s + 2.37·7-s + 2.82i·8-s + 2.99·9-s − 9.64i·10-s + 7.57i·11-s + 3.46·12-s + 19.3i·13-s − 3.35i·14-s − 11.8·15-s + 4.00·16-s − 21.1·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s − 0.500·4-s + 1.36·5-s + 0.408i·6-s + 0.338·7-s + 0.353i·8-s + 0.333·9-s − 0.964i·10-s + 0.688i·11-s + 0.288·12-s + 1.48i·13-s − 0.239i·14-s − 0.787·15-s + 0.250·16-s − 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.975 + 0.221i$
Analytic conductor: \(9.64580\)
Root analytic conductor: \(3.10576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1),\ 0.975 + 0.221i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.69608 - 0.189813i\)
\(L(\frac12)\) \(\approx\) \(1.69608 - 0.189813i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + 1.73T \)
59 \( 1 + (-13.0 + 57.5i)T \)
good5 \( 1 - 6.82T + 25T^{2} \)
7 \( 1 - 2.37T + 49T^{2} \)
11 \( 1 - 7.57iT - 121T^{2} \)
13 \( 1 - 19.3iT - 169T^{2} \)
17 \( 1 + 21.1T + 289T^{2} \)
19 \( 1 - 33.6T + 361T^{2} \)
23 \( 1 - 13.4iT - 529T^{2} \)
29 \( 1 - 42.0T + 841T^{2} \)
31 \( 1 + 35.8iT - 961T^{2} \)
37 \( 1 - 39.6iT - 1.36e3T^{2} \)
41 \( 1 - 63.8T + 1.68e3T^{2} \)
43 \( 1 + 7.54iT - 1.84e3T^{2} \)
47 \( 1 + 73.6iT - 2.20e3T^{2} \)
53 \( 1 - 51.3T + 2.80e3T^{2} \)
61 \( 1 - 108. iT - 3.72e3T^{2} \)
67 \( 1 + 78.9iT - 4.48e3T^{2} \)
71 \( 1 + 89.3T + 5.04e3T^{2} \)
73 \( 1 - 88.7iT - 5.32e3T^{2} \)
79 \( 1 + 97.3T + 6.24e3T^{2} \)
83 \( 1 - 87.0iT - 6.88e3T^{2} \)
89 \( 1 + 145. iT - 7.92e3T^{2} \)
97 \( 1 - 5.99iT - 9.40e3T^{2} \)
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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

−11.45782822141853989554546591137, −10.16472776513120947923209611335, −9.653629252250109607812722311790, −8.848327878518456658624457546050, −7.24869096223605416514543344941, −6.26406430656272667513647165801, −5.18179975445062590833203671783, −4.29389091398023095660637740715, −2.41454819581949601428003140996, −1.41786482534891901780618625104, 0.980757052139986659047611785253, 2.87041458129109598921913592102, 4.74574663684856241549089183048, 5.61661636768918401949698838460, 6.20368173747671080169123828214, 7.34986505922397946149860866718, 8.474950122316569275099205476078, 9.424945540626775309786362700565, 10.33794707472719482785075348676, 11.05828902062820915727985312888

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