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Properties

Label 2-354-177.11-c1-0-2
Degree $2$
Conductor $354$
Sign $-0.804 - 0.593i$
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  + (−0.976 + 0.214i)2-s + (−0.233 + 1.71i)3-s + (0.907 − 0.419i)4-s + (2.85 + 0.792i)5-s + (−0.140 − 1.72i)6-s + (−3.45 + 3.27i)7-s + (−0.796 + 0.605i)8-s + (−2.89 − 0.801i)9-s + (−2.95 − 0.160i)10-s + (−0.467 − 0.549i)11-s + (0.508 + 1.65i)12-s + (0.999 + 2.96i)13-s + (2.67 − 3.94i)14-s + (−2.02 + 4.71i)15-s + (0.647 − 0.762i)16-s + (3.74 − 3.95i)17-s + ⋯
L(s)  = 1  + (−0.690 + 0.152i)2-s + (−0.134 + 0.990i)3-s + (0.453 − 0.209i)4-s + (1.27 + 0.354i)5-s + (−0.0574 − 0.704i)6-s + (−1.30 + 1.23i)7-s + (−0.281 + 0.213i)8-s + (−0.963 − 0.267i)9-s + (−0.934 − 0.0506i)10-s + (−0.140 − 0.165i)11-s + (0.146 + 0.477i)12-s + (0.277 + 0.822i)13-s + (0.714 − 1.05i)14-s + (−0.523 + 1.21i)15-s + (0.161 − 0.190i)16-s + (0.907 − 0.958i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.804 - 0.593i$
Analytic conductor: \(2.82670\)
Root analytic conductor: \(1.68128\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :1/2),\ -0.804 - 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.266401 + 0.809519i\)
\(L(\frac12)\) \(\approx\) \(0.266401 + 0.809519i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.976 - 0.214i)T \)
3 \( 1 + (0.233 - 1.71i)T \)
59 \( 1 + (3.58 + 6.79i)T \)
good5 \( 1 + (-2.85 - 0.792i)T + (4.28 + 2.57i)T^{2} \)
7 \( 1 + (3.45 - 3.27i)T + (0.378 - 6.98i)T^{2} \)
11 \( 1 + (0.467 + 0.549i)T + (-1.77 + 10.8i)T^{2} \)
13 \( 1 + (-0.999 - 2.96i)T + (-10.3 + 7.86i)T^{2} \)
17 \( 1 + (-3.74 + 3.95i)T + (-0.920 - 16.9i)T^{2} \)
19 \( 1 + (-0.338 - 0.849i)T + (-13.7 + 13.0i)T^{2} \)
23 \( 1 + (5.93 - 0.645i)T + (22.4 - 4.94i)T^{2} \)
29 \( 1 + (1.76 - 8.01i)T + (-26.3 - 12.1i)T^{2} \)
31 \( 1 + (1.16 + 0.464i)T + (22.5 + 21.3i)T^{2} \)
37 \( 1 + (4.11 - 5.41i)T + (-9.89 - 35.6i)T^{2} \)
41 \( 1 + (-0.819 + 7.53i)T + (-40.0 - 8.81i)T^{2} \)
43 \( 1 + (-5.58 - 4.74i)T + (6.95 + 42.4i)T^{2} \)
47 \( 1 + (-0.560 - 2.01i)T + (-40.2 + 24.2i)T^{2} \)
53 \( 1 + (-7.64 + 0.414i)T + (52.6 - 5.73i)T^{2} \)
61 \( 1 + (-1.38 - 6.27i)T + (-55.3 + 25.6i)T^{2} \)
67 \( 1 + (-7.40 - 9.74i)T + (-17.9 + 64.5i)T^{2} \)
71 \( 1 + (-6.08 + 1.68i)T + (60.8 - 36.6i)T^{2} \)
73 \( 1 + (-8.57 - 5.81i)T + (27.0 + 67.8i)T^{2} \)
79 \( 1 + (1.47 + 9.01i)T + (-74.8 + 25.2i)T^{2} \)
83 \( 1 + (-4.67 - 8.82i)T + (-46.5 + 68.6i)T^{2} \)
89 \( 1 + (4.59 + 1.01i)T + (80.7 + 37.3i)T^{2} \)
97 \( 1 + (4.29 - 2.91i)T + (35.9 - 90.1i)T^{2} \)
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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

−11.67179004796649857169152674834, −10.52390657796692046036548850749, −9.715172649920825971746507417185, −9.411169076029689256441231434664, −8.577284556784038451178122450993, −6.84917862563854782643738286214, −5.92166800395477691138563391933, −5.42297113371398386371209092154, −3.38754441078091562052525444574, −2.32743410739604941313044929781, 0.72082597016399143950580962298, 2.13039153085776790573539233835, 3.58818958667236196793623877688, 5.76141130422780599795603564368, 6.26843552528425040037738155323, 7.36862997365077848850945754308, 8.181682705830209186822782460176, 9.468428216016596348813311426270, 10.11325356180378315637851619898, 10.78036245383329759500358892066

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