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Properties

Label 2-354-177.11-c1-0-7
Degree $2$
Conductor $354$
Sign $0.918 - 0.396i$
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 + (−1.36 + 1.07i)3-s + (0.907 − 0.419i)4-s + (1.88 + 0.524i)5-s + (−1.09 + 1.33i)6-s + (0.743 − 0.704i)7-s + (0.796 − 0.605i)8-s + (0.705 − 2.91i)9-s + (1.95 + 0.106i)10-s + (−0.616 − 0.726i)11-s + (−0.785 + 1.54i)12-s + (2.28 + 6.77i)13-s + (0.574 − 0.847i)14-s + (−3.13 + 1.31i)15-s + (0.647 − 0.762i)16-s + (2.91 − 3.07i)17-s + ⋯
L(s)  = 1  + (0.690 − 0.152i)2-s + (−0.785 + 0.618i)3-s + (0.453 − 0.209i)4-s + (0.845 + 0.234i)5-s + (−0.448 + 0.546i)6-s + (0.280 − 0.266i)7-s + (0.281 − 0.213i)8-s + (0.235 − 0.971i)9-s + (0.619 + 0.0335i)10-s + (−0.185 − 0.218i)11-s + (−0.226 + 0.445i)12-s + (0.633 + 1.87i)13-s + (0.153 − 0.226i)14-s + (−0.809 + 0.338i)15-s + (0.161 − 0.190i)16-s + (0.706 − 0.745i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.918 - 0.396i$
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: \((2,\ 354,\ (\ :1/2),\ 0.918 - 0.396i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82027 + 0.376261i\)
\(L(\frac12)\) \(\approx\) \(1.82027 + 0.376261i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 - 1.07i)T \)
59 \( 1 + (5.74 - 5.10i)T \)
good5 \( 1 + (-1.88 - 0.524i)T + (4.28 + 2.57i)T^{2} \)
7 \( 1 + (-0.743 + 0.704i)T + (0.378 - 6.98i)T^{2} \)
11 \( 1 + (0.616 + 0.726i)T + (-1.77 + 10.8i)T^{2} \)
13 \( 1 + (-2.28 - 6.77i)T + (-10.3 + 7.86i)T^{2} \)
17 \( 1 + (-2.91 + 3.07i)T + (-0.920 - 16.9i)T^{2} \)
19 \( 1 + (-1.60 - 4.03i)T + (-13.7 + 13.0i)T^{2} \)
23 \( 1 + (1.40 - 0.152i)T + (22.4 - 4.94i)T^{2} \)
29 \( 1 + (0.263 - 1.19i)T + (-26.3 - 12.1i)T^{2} \)
31 \( 1 + (4.72 + 1.88i)T + (22.5 + 21.3i)T^{2} \)
37 \( 1 + (-5.87 + 7.72i)T + (-9.89 - 35.6i)T^{2} \)
41 \( 1 + (0.0278 - 0.256i)T + (-40.0 - 8.81i)T^{2} \)
43 \( 1 + (1.05 + 0.892i)T + (6.95 + 42.4i)T^{2} \)
47 \( 1 + (3.52 + 12.6i)T + (-40.2 + 24.2i)T^{2} \)
53 \( 1 + (8.90 - 0.482i)T + (52.6 - 5.73i)T^{2} \)
61 \( 1 + (1.02 + 4.63i)T + (-55.3 + 25.6i)T^{2} \)
67 \( 1 + (5.80 + 7.63i)T + (-17.9 + 64.5i)T^{2} \)
71 \( 1 + (8.61 - 2.39i)T + (60.8 - 36.6i)T^{2} \)
73 \( 1 + (0.629 + 0.427i)T + (27.0 + 67.8i)T^{2} \)
79 \( 1 + (0.201 + 1.23i)T + (-74.8 + 25.2i)T^{2} \)
83 \( 1 + (2.69 + 5.07i)T + (-46.5 + 68.6i)T^{2} \)
89 \( 1 + (1.01 + 0.223i)T + (80.7 + 37.3i)T^{2} \)
97 \( 1 + (-8.25 + 5.59i)T + (35.9 - 90.1i)T^{2} \)
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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

−11.52462348537242482653590607729, −10.77981537537541361311154085934, −9.853374515497682597505015269584, −9.150316253185344899959361503748, −7.43398362493533203689631509202, −6.30509755648615929680630959996, −5.68586645989062931239105675016, −4.56127327219605786621960332593, −3.56718392151735118134096107386, −1.74243737021901734262836336430, 1.46003545513910639964436274636, 2.99091150433769050338941991825, 4.81900841536365903227671931026, 5.67206254977751779739938609025, 6.16324425484432364279783947522, 7.53412176389895864526315636444, 8.272758956457721096435483917802, 9.827206691332136138682754938733, 10.70326437787978362821021733387, 11.50790718515104999877005437045

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