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Properties

Label 2-354-3.2-c2-0-0
Degree $2$
Conductor $354$
Sign $-0.999 - 0.0131i$
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 + (−2.99 − 0.0395i)3-s − 2.00·4-s − 8.98i·5-s + (0.0559 − 4.24i)6-s + 1.46·7-s − 2.82i·8-s + (8.99 + 0.237i)9-s + 12.7·10-s + 2.04i·11-s + (5.99 + 0.0791i)12-s − 17.4·13-s + 2.06i·14-s + (−0.355 + 26.9i)15-s + 4.00·16-s + 30.2i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.999 − 0.0131i)3-s − 0.500·4-s − 1.79i·5-s + (0.00933 − 0.707i)6-s + 0.208·7-s − 0.353i·8-s + (0.999 + 0.0263i)9-s + 1.27·10-s + 0.185i·11-s + (0.499 + 0.00659i)12-s − 1.34·13-s + 0.147i·14-s + (−0.0237 + 1.79i)15-s + 0.250·16-s + 1.77i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0131i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.999 - 0.0131i$
Analytic conductor: \(9.64580\)
Root analytic conductor: \(3.10576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1),\ -0.999 - 0.0131i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.000136160 + 0.0206333i\)
\(L(\frac12)\) \(\approx\) \(0.000136160 + 0.0206333i\)
\(L(2)\) 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 \( 1 + (2.99 + 0.0395i)T \)
59 \( 1 - 7.68iT \)
good5 \( 1 + 8.98iT - 25T^{2} \)
7 \( 1 - 1.46T + 49T^{2} \)
11 \( 1 - 2.04iT - 121T^{2} \)
13 \( 1 + 17.4T + 169T^{2} \)
17 \( 1 - 30.2iT - 289T^{2} \)
19 \( 1 + 8.78T + 361T^{2} \)
23 \( 1 + 44.5iT - 529T^{2} \)
29 \( 1 - 38.9iT - 841T^{2} \)
31 \( 1 - 24.6T + 961T^{2} \)
37 \( 1 + 51.2T + 1.36e3T^{2} \)
41 \( 1 - 46.9iT - 1.68e3T^{2} \)
43 \( 1 - 11.8T + 1.84e3T^{2} \)
47 \( 1 - 27.9iT - 2.20e3T^{2} \)
53 \( 1 + 12.8iT - 2.80e3T^{2} \)
61 \( 1 + 42.0T + 3.72e3T^{2} \)
67 \( 1 + 101.T + 4.48e3T^{2} \)
71 \( 1 - 62.5iT - 5.04e3T^{2} \)
73 \( 1 - 52.7T + 5.32e3T^{2} \)
79 \( 1 + 85.5T + 6.24e3T^{2} \)
83 \( 1 - 16.6iT - 6.88e3T^{2} \)
89 \( 1 + 1.71iT - 7.92e3T^{2} \)
97 \( 1 + 146.T + 9.40e3T^{2} \)
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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

−12.15856999777242770251307238092, −10.66422239047739517117363074663, −9.830407132679041701518557734998, −8.715659255807911875985954826418, −8.059565303123628875596746517472, −6.78745746981238769696248527797, −5.79022676462563295584915256453, −4.78108791803670932456550097121, −4.41147657803842268822736670958, −1.49704922552949171117600866613, 0.01081335997477785838301867184, 2.20425059247802451987014909371, 3.36054549452656121902986574787, 4.76392881974265357473335910986, 5.84528310059121357498260716655, 7.06189036040746954099382683984, 7.54751441968035879868211988480, 9.529757863699168356841636810748, 10.08316571373136366717729514427, 10.91153146352492376650433044113

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