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Properties

Label 2-354-3.2-c2-0-24
Degree $2$
Conductor $354$
Sign $-0.496 + 0.868i$
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 + (−1.48 + 2.60i)3-s − 2.00·4-s + 4.30i·5-s + (3.68 + 2.10i)6-s − 7.55·7-s + 2.82i·8-s + (−4.56 − 7.75i)9-s + 6.09·10-s − 1.44i·11-s + (2.97 − 5.20i)12-s + 1.74·13-s + 10.6i·14-s + (−11.2 − 6.41i)15-s + 4.00·16-s − 8.57i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.496 + 0.868i)3-s − 0.500·4-s + 0.861i·5-s + (0.613 + 0.351i)6-s − 1.07·7-s + 0.353i·8-s + (−0.507 − 0.861i)9-s + 0.609·10-s − 0.130i·11-s + (0.248 − 0.434i)12-s + 0.134·13-s + 0.763i·14-s + (−0.747 − 0.427i)15-s + 0.250·16-s − 0.504i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.496 + 0.868i$
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: \((2,\ 354,\ (\ :1),\ -0.496 + 0.868i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.231939 - 0.399838i\)
\(L(\frac12)\) \(\approx\) \(0.231939 - 0.399838i\)
\(L(2)\) 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 \( 1 + (1.48 - 2.60i)T \)
59 \( 1 + 7.68iT \)
good5 \( 1 - 4.30iT - 25T^{2} \)
7 \( 1 + 7.55T + 49T^{2} \)
11 \( 1 + 1.44iT - 121T^{2} \)
13 \( 1 - 1.74T + 169T^{2} \)
17 \( 1 + 8.57iT - 289T^{2} \)
19 \( 1 - 8.46T + 361T^{2} \)
23 \( 1 + 38.3iT - 529T^{2} \)
29 \( 1 + 41.9iT - 841T^{2} \)
31 \( 1 + 28.6T + 961T^{2} \)
37 \( 1 - 13.6T + 1.36e3T^{2} \)
41 \( 1 - 32.2iT - 1.68e3T^{2} \)
43 \( 1 + 82.1T + 1.84e3T^{2} \)
47 \( 1 - 18.5iT - 2.20e3T^{2} \)
53 \( 1 + 84.8iT - 2.80e3T^{2} \)
61 \( 1 + 14.2T + 3.72e3T^{2} \)
67 \( 1 + 35.0T + 4.48e3T^{2} \)
71 \( 1 - 24.2iT - 5.04e3T^{2} \)
73 \( 1 - 10.7T + 5.32e3T^{2} \)
79 \( 1 - 5.01T + 6.24e3T^{2} \)
83 \( 1 + 51.3iT - 6.88e3T^{2} \)
89 \( 1 - 62.4iT - 7.92e3T^{2} \)
97 \( 1 - 22.0T + 9.40e3T^{2} \)
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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

−10.92719226253468829708330367907, −10.03335682669470835061573138297, −9.601820247603564852756447487865, −8.443239997193699012709904281305, −6.83824305111559195889508073615, −6.05949883844709071612420115678, −4.75356659610231112348748392801, −3.55645460263571471576976745794, −2.75631855420265106818353170859, −0.23809881053392382272752413794, 1.36938449215783045089002031462, 3.43356529196673012665384402104, 5.04389694482571902952124995578, 5.79878771414882658687463166070, 6.80287021795945536211499336218, 7.57764208580673678074551519018, 8.679398782689135003826722619918, 9.448728693197334713055355625558, 10.62422227443927912745747170229, 11.85529766640695581820056660653

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