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Properties

Label 2-354-177.11-c1-0-13
Degree $2$
Conductor $354$
Sign $0.808 - 0.588i$
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 + (0.566 + 1.63i)3-s + (0.907 − 0.419i)4-s + (3.87 + 1.07i)5-s + (0.904 + 1.47i)6-s + (−0.391 + 0.370i)7-s + (0.796 − 0.605i)8-s + (−2.35 + 1.85i)9-s + (4.01 + 0.217i)10-s + (−3.75 − 4.42i)11-s + (1.20 + 1.24i)12-s + (−1.27 − 3.79i)13-s + (−0.302 + 0.446i)14-s + (0.433 + 6.94i)15-s + (0.647 − 0.762i)16-s + (−4.09 + 4.32i)17-s + ⋯
L(s)  = 1  + (0.690 − 0.152i)2-s + (0.326 + 0.945i)3-s + (0.453 − 0.209i)4-s + (1.73 + 0.480i)5-s + (0.369 + 0.602i)6-s + (−0.147 + 0.140i)7-s + (0.281 − 0.213i)8-s + (−0.786 + 0.618i)9-s + (1.26 + 0.0688i)10-s + (−1.13 − 1.33i)11-s + (0.346 + 0.360i)12-s + (−0.354 − 1.05i)13-s + (−0.0808 + 0.119i)14-s + (0.111 + 1.79i)15-s + (0.161 − 0.190i)16-s + (−0.993 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.808 - 0.588i$
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: \((2,\ 354,\ (\ :1/2),\ 0.808 - 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38466 + 0.776124i\)
\(L(\frac12)\) \(\approx\) \(2.38466 + 0.776124i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.566 - 1.63i)T \)
59 \( 1 + (7.36 + 2.18i)T \)
good5 \( 1 + (-3.87 - 1.07i)T + (4.28 + 2.57i)T^{2} \)
7 \( 1 + (0.391 - 0.370i)T + (0.378 - 6.98i)T^{2} \)
11 \( 1 + (3.75 + 4.42i)T + (-1.77 + 10.8i)T^{2} \)
13 \( 1 + (1.27 + 3.79i)T + (-10.3 + 7.86i)T^{2} \)
17 \( 1 + (4.09 - 4.32i)T + (-0.920 - 16.9i)T^{2} \)
19 \( 1 + (0.599 + 1.50i)T + (-13.7 + 13.0i)T^{2} \)
23 \( 1 + (1.60 - 0.174i)T + (22.4 - 4.94i)T^{2} \)
29 \( 1 + (-0.936 + 4.25i)T + (-26.3 - 12.1i)T^{2} \)
31 \( 1 + (-2.54 - 1.01i)T + (22.5 + 21.3i)T^{2} \)
37 \( 1 + (-6.03 + 7.93i)T + (-9.89 - 35.6i)T^{2} \)
41 \( 1 + (1.21 - 11.1i)T + (-40.0 - 8.81i)T^{2} \)
43 \( 1 + (2.32 + 1.97i)T + (6.95 + 42.4i)T^{2} \)
47 \( 1 + (-2.73 - 9.86i)T + (-40.2 + 24.2i)T^{2} \)
53 \( 1 + (-7.26 + 0.393i)T + (52.6 - 5.73i)T^{2} \)
61 \( 1 + (-0.853 - 3.87i)T + (-55.3 + 25.6i)T^{2} \)
67 \( 1 + (1.76 + 2.31i)T + (-17.9 + 64.5i)T^{2} \)
71 \( 1 + (4.82 - 1.34i)T + (60.8 - 36.6i)T^{2} \)
73 \( 1 + (1.97 + 1.33i)T + (27.0 + 67.8i)T^{2} \)
79 \( 1 + (2.27 + 13.8i)T + (-74.8 + 25.2i)T^{2} \)
83 \( 1 + (-1.60 - 3.03i)T + (-46.5 + 68.6i)T^{2} \)
89 \( 1 + (5.24 + 1.15i)T + (80.7 + 37.3i)T^{2} \)
97 \( 1 + (-2.74 + 1.85i)T + (35.9 - 90.1i)T^{2} \)
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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

−11.15780497793417301963860197556, −10.58988918224493758086741015270, −10.01983622543760274247689633464, −8.999513241484610646951049274212, −7.920533893240187495143369919005, −6.03012938166198608491401696536, −5.86405497549240877299833770320, −4.66504118451380812914612604831, −3.04886793530849116088155518615, −2.43648734176414583087377134652, 1.91620792851584401778865579371, 2.52846460229061303063736707158, 4.64283065471060231202527032462, 5.50567965060549861759462819999, 6.64660783522850085274558488493, 7.18199371161508873355941016517, 8.582320145813026075503480617194, 9.532661507842655252705268713998, 10.31872626229527924157631765223, 11.79438804801971123038932690100

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