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Properties

Label 2-354-177.176-c1-0-3
Degree $2$
Conductor $354$
Sign $-0.247 - 0.968i$
Analytic cond. $2.82670$
Root an. cond. $1.68128$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (0.0738 + 1.73i)3-s + 4-s + 2.30i·5-s + (−0.0738 − 1.73i)6-s + 3.25·7-s − 8-s + (−2.98 + 0.255i)9-s − 2.30i·10-s + 4.56·11-s + (0.0738 + 1.73i)12-s + 2.44i·13-s − 3.25·14-s + (−3.98 + 0.170i)15-s + 16-s − 2.52i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.0426 + 0.999i)3-s + 0.5·4-s + 1.03i·5-s + (−0.0301 − 0.706i)6-s + 1.22·7-s − 0.353·8-s + (−0.996 + 0.0851i)9-s − 0.728i·10-s + 1.37·11-s + (0.0213 + 0.499i)12-s + 0.677i·13-s − 0.869·14-s + (−1.02 + 0.0439i)15-s + 0.250·16-s − 0.613i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.247 - 0.968i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ -0.247 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.671038 + 0.863923i\)
\(L(\frac12)\) \(\approx\) \(0.671038 + 0.863923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.0738 - 1.73i)T \)
59 \( 1 + (-7.51 + 1.58i)T \)
good5 \( 1 - 2.30iT - 5T^{2} \)
7 \( 1 - 3.25T + 7T^{2} \)
11 \( 1 - 4.56T + 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + 2.52iT - 17T^{2} \)
19 \( 1 + 1.31T + 19T^{2} \)
23 \( 1 + 8.45T + 23T^{2} \)
29 \( 1 - 7.58iT - 29T^{2} \)
31 \( 1 + 1.81iT - 31T^{2} \)
37 \( 1 + 2.83iT - 37T^{2} \)
41 \( 1 + 5.77iT - 41T^{2} \)
43 \( 1 - 11.5iT - 43T^{2} \)
47 \( 1 - 0.674T + 47T^{2} \)
53 \( 1 + 8.22iT - 53T^{2} \)
61 \( 1 + 0.133iT - 61T^{2} \)
67 \( 1 - 3.33iT - 67T^{2} \)
71 \( 1 + 6.35iT - 71T^{2} \)
73 \( 1 + 13.4iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 9.52T + 83T^{2} \)
89 \( 1 - 4.28T + 89T^{2} \)
97 \( 1 + 8.58iT - 97T^{2} \)
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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

−11.38345012159794717105581159009, −10.84452985732150396477765132306, −9.885065356662206620869364684888, −9.070450363969016997405637630344, −8.201073368424374997376637712734, −7.06215379438428731673182071306, −6.07298102107745027774432939384, −4.64291705430241087043772374442, −3.54850221766906354572284753009, −2.00777769652594529976108254401, 1.04306865061571909889204037014, 2.02671837181188182544339155016, 4.11011229424872697636413721934, 5.54151165007494036233098631018, 6.49640449212890766709951904877, 7.79590360237408163983920673065, 8.317904832718459719458845950778, 8.989955220328544475247476141497, 10.23677226276624467466012370570, 11.48979797641064989346758236983

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