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Properties

Label 2-354-59.58-c4-0-36
Degree $2$
Conductor $354$
Sign $0.120 + 0.992i$
Analytic cond. $36.5929$
Root an. cond. $6.04921$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s − 5.19·3-s − 8.00·4-s + 30.0·5-s − 14.6i·6-s + 18.5·7-s − 22.6i·8-s + 27·9-s + 85.0i·10-s − 116. i·11-s + 41.5·12-s − 225. i·13-s + 52.5i·14-s − 156.·15-s + 64.0·16-s − 75.8·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577·3-s − 0.500·4-s + 1.20·5-s − 0.408i·6-s + 0.379·7-s − 0.353i·8-s + 0.333·9-s + 0.850i·10-s − 0.962i·11-s + 0.288·12-s − 1.33i·13-s + 0.268i·14-s − 0.694·15-s + 0.250·16-s − 0.262·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.120 + 0.992i$
Analytic conductor: \(36.5929\)
Root analytic conductor: \(6.04921\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :2),\ 0.120 + 0.992i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.030364944\)
\(L(\frac12)\) \(\approx\) \(1.030364944\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 + 5.19T \)
59 \( 1 + (-3.45e3 + 419. i)T \)
good5 \( 1 - 30.0T + 625T^{2} \)
7 \( 1 - 18.5T + 2.40e3T^{2} \)
11 \( 1 + 116. iT - 1.46e4T^{2} \)
13 \( 1 + 225. iT - 2.85e4T^{2} \)
17 \( 1 + 75.8T + 8.35e4T^{2} \)
19 \( 1 + 326.T + 1.30e5T^{2} \)
23 \( 1 - 486. iT - 2.79e5T^{2} \)
29 \( 1 + 1.49e3T + 7.07e5T^{2} \)
31 \( 1 - 323. iT - 9.23e5T^{2} \)
37 \( 1 + 1.38e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.35e3T + 2.82e6T^{2} \)
43 \( 1 + 762. iT - 3.41e6T^{2} \)
47 \( 1 + 1.04e3iT - 4.87e6T^{2} \)
53 \( 1 + 20.6T + 7.89e6T^{2} \)
61 \( 1 + 2.47e3iT - 1.38e7T^{2} \)
67 \( 1 + 7.90e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.91e3T + 2.54e7T^{2} \)
73 \( 1 - 2.89e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.60e3T + 3.89e7T^{2} \)
83 \( 1 + 1.77e3iT - 4.74e7T^{2} \)
89 \( 1 - 3.55e3iT - 6.27e7T^{2} \)
97 \( 1 + 735. iT - 8.85e7T^{2} \)
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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

−10.57402015220582889169437290248, −9.693362372096414392372988437770, −8.707829423583366684602473722904, −7.73758065088287528480611193652, −6.54540350365565944049366652198, −5.65525163961153027883109005066, −5.22996354466429444860386709898, −3.57731572727704144629062791292, −1.83537686143723206996081182263, −0.30576678949989009326650389897, 1.59733847041102292738594293611, 2.22685642894327087316470480595, 4.16495527486262061011811726208, 4.99267287438107538112415247906, 6.15235202177584440692904491214, 7.03533470975114962901949121086, 8.557676183073805590500854359511, 9.541940642865118526743504078140, 10.10082911689713651046297895533, 11.09383043938426474245586838514

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