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Properties

Label 2-354-3.2-c2-0-21
Degree $2$
Conductor $354$
Sign $0.891 - 0.452i$
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 + (−2.67 + 1.35i)3-s − 2.00·4-s + 1.30i·5-s + (−1.92 − 3.78i)6-s + 6.99·7-s − 2.82i·8-s + (5.31 − 7.26i)9-s − 1.84·10-s − 17.0i·11-s + (5.35 − 2.71i)12-s − 9.38·13-s + 9.89i·14-s + (−1.77 − 3.48i)15-s + 4.00·16-s − 16.9i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.891 + 0.452i)3-s − 0.500·4-s + 0.260i·5-s + (−0.320 − 0.630i)6-s + 0.999·7-s − 0.353i·8-s + (0.590 − 0.807i)9-s − 0.184·10-s − 1.54i·11-s + (0.445 − 0.226i)12-s − 0.721·13-s + 0.706i·14-s + (−0.118 − 0.232i)15-s + 0.250·16-s − 0.994i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.891 - 0.452i$
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: \((2,\ 354,\ (\ :1),\ 0.891 - 0.452i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19334 + 0.285553i\)
\(L(\frac12)\) \(\approx\) \(1.19334 + 0.285553i\)
\(L(2)\) 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 \( 1 + (2.67 - 1.35i)T \)
59 \( 1 + 7.68iT \)
good5 \( 1 - 1.30iT - 25T^{2} \)
7 \( 1 - 6.99T + 49T^{2} \)
11 \( 1 + 17.0iT - 121T^{2} \)
13 \( 1 + 9.38T + 169T^{2} \)
17 \( 1 + 16.9iT - 289T^{2} \)
19 \( 1 - 13.0T + 361T^{2} \)
23 \( 1 - 4.64iT - 529T^{2} \)
29 \( 1 - 39.4iT - 841T^{2} \)
31 \( 1 - 41.0T + 961T^{2} \)
37 \( 1 - 65.8T + 1.36e3T^{2} \)
41 \( 1 + 77.9iT - 1.68e3T^{2} \)
43 \( 1 + 74.0T + 1.84e3T^{2} \)
47 \( 1 - 51.6iT - 2.20e3T^{2} \)
53 \( 1 + 78.0iT - 2.80e3T^{2} \)
61 \( 1 - 73.8T + 3.72e3T^{2} \)
67 \( 1 + 87.3T + 4.48e3T^{2} \)
71 \( 1 + 31.0iT - 5.04e3T^{2} \)
73 \( 1 + 14.4T + 5.32e3T^{2} \)
79 \( 1 - 109.T + 6.24e3T^{2} \)
83 \( 1 - 21.4iT - 6.88e3T^{2} \)
89 \( 1 + 37.4iT - 7.92e3T^{2} \)
97 \( 1 - 167.T + 9.40e3T^{2} \)
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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

−11.34011758475512989428385165242, −10.49864942918493247237223438029, −9.440758565590534745124138247836, −8.462127374000936315500324954685, −7.39227705184845358116073244113, −6.43930291327944133323297942659, −5.33010584547757050042767536989, −4.77798200126685272721661749643, −3.25109125425319994385293796018, −0.78247002947376980693253887195, 1.19622034545417256046889668383, 2.31112338765973884323025585513, 4.52543833063413448478305887319, 4.86197713847241484514057364730, 6.27409031823097605658563358055, 7.51269967615629128096303465593, 8.237488860003067591898857848345, 9.763798986313080100738433022165, 10.28379661515539875024803397947, 11.48381366568013702139732299261

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