Properties

Label 2-354-3.2-c2-0-6
Degree $2$
Conductor $354$
Sign $-0.410 + 0.911i$
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 + (−1.23 + 2.73i)3-s − 2.00·4-s + 5.41i·5-s + (−3.86 − 1.74i)6-s + 0.567·7-s − 2.82i·8-s + (−5.97 − 6.73i)9-s − 7.65·10-s + 16.7i·11-s + (2.46 − 5.47i)12-s − 7.52·13-s + 0.802i·14-s + (−14.8 − 6.66i)15-s + 4.00·16-s + 2.38i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.410 + 0.911i)3-s − 0.500·4-s + 1.08i·5-s + (−0.644 − 0.290i)6-s + 0.0811·7-s − 0.353i·8-s + (−0.663 − 0.748i)9-s − 0.765·10-s + 1.52i·11-s + (0.205 − 0.455i)12-s − 0.578·13-s + 0.0573i·14-s + (−0.987 − 0.444i)15-s + 0.250·16-s + 0.140i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.410 + 0.911i$
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: \((2,\ 354,\ (\ :1),\ -0.410 + 0.911i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.395124 - 0.610999i\)
\(L(\frac12)\) \(\approx\) \(0.395124 - 0.610999i\)
\(L(2)\) 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 \( 1 + (1.23 - 2.73i)T \)
59 \( 1 - 7.68iT \)
good5 \( 1 - 5.41iT - 25T^{2} \)
7 \( 1 - 0.567T + 49T^{2} \)
11 \( 1 - 16.7iT - 121T^{2} \)
13 \( 1 + 7.52T + 169T^{2} \)
17 \( 1 - 2.38iT - 289T^{2} \)
19 \( 1 + 18.4T + 361T^{2} \)
23 \( 1 + 17.2iT - 529T^{2} \)
29 \( 1 + 18.4iT - 841T^{2} \)
31 \( 1 - 47.0T + 961T^{2} \)
37 \( 1 + 0.903T + 1.36e3T^{2} \)
41 \( 1 + 38.6iT - 1.68e3T^{2} \)
43 \( 1 + 43.2T + 1.84e3T^{2} \)
47 \( 1 - 15.2iT - 2.20e3T^{2} \)
53 \( 1 - 93.8iT - 2.80e3T^{2} \)
61 \( 1 + 44.5T + 3.72e3T^{2} \)
67 \( 1 - 50.8T + 4.48e3T^{2} \)
71 \( 1 + 2.51iT - 5.04e3T^{2} \)
73 \( 1 + 16.9T + 5.32e3T^{2} \)
79 \( 1 + 34.1T + 6.24e3T^{2} \)
83 \( 1 - 9.89iT - 6.88e3T^{2} \)
89 \( 1 - 31.7iT - 7.92e3T^{2} \)
97 \( 1 + 96.7T + 9.40e3T^{2} \)
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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

−11.83633378359782236421611525146, −10.60411251336954581768674132502, −10.15315402678162679471892819531, −9.256977358027414532102734128407, −8.033106695601015102727983836715, −6.89464412974340012040279135355, −6.27684811775699132007644289809, −4.90390180375996382023532141116, −4.16208800993626174860462105167, −2.62221270927383083351436803958, 0.35559889916815797543731587845, 1.53021907903984536406010514364, 3.05001603092697948854805999940, 4.68989347528505154691897850604, 5.55764893990266387559995746243, 6.66684733680789227916438150212, 8.208741158045787377980649915462, 8.506646769257057103384590597001, 9.749272294949165583001026640847, 10.95063346172096483065696920163

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