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Properties

Label 2-1911-1911.356-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.185 - 0.982i$
Analytic cond. $0.953713$
Root an. cond. $0.976582$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.781 + 0.623i)3-s + (0.433 + 0.900i)4-s + (−0.222 − 0.974i)7-s + (0.222 − 0.974i)9-s + (−0.900 − 0.433i)12-s + (0.781 + 0.623i)13-s + (−0.623 + 0.781i)16-s + (1.19 + 1.19i)19-s + (0.781 + 0.623i)21-s + (−0.974 − 0.222i)25-s + (0.433 + 0.900i)27-s + (0.781 − 0.623i)28-s + (1.33 + 1.33i)31-s + (0.974 − 0.222i)36-s + (−0.623 − 1.78i)37-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)3-s + (0.433 + 0.900i)4-s + (−0.222 − 0.974i)7-s + (0.222 − 0.974i)9-s + (−0.900 − 0.433i)12-s + (0.781 + 0.623i)13-s + (−0.623 + 0.781i)16-s + (1.19 + 1.19i)19-s + (0.781 + 0.623i)21-s + (−0.974 − 0.222i)25-s + (0.433 + 0.900i)27-s + (0.781 − 0.623i)28-s + (1.33 + 1.33i)31-s + (0.974 − 0.222i)36-s + (−0.623 − 1.78i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.185 - 0.982i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.185 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9766916575\)
\(L(\frac12)\) \(\approx\) \(0.9766916575\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.781 - 0.623i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.781 - 0.623i)T \)
good2 \( 1 + (-0.433 - 0.900i)T^{2} \)
5 \( 1 + (0.974 + 0.222i)T^{2} \)
11 \( 1 + (0.433 + 0.900i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 + (-1.19 - 1.19i)T + iT^{2} \)
23 \( 1 + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (-1.33 - 1.33i)T + iT^{2} \)
37 \( 1 + (0.623 + 1.78i)T + (-0.781 + 0.623i)T^{2} \)
41 \( 1 + (0.974 + 0.222i)T^{2} \)
43 \( 1 + (-1.52 - 1.21i)T + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.433 - 0.900i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 + (0.974 - 0.222i)T^{2} \)
61 \( 1 + (0.781 - 1.62i)T + (-0.623 - 0.781i)T^{2} \)
67 \( 1 + (-0.158 + 0.158i)T - iT^{2} \)
71 \( 1 + (0.781 + 0.623i)T^{2} \)
73 \( 1 + (0.351 + 0.559i)T + (-0.433 + 0.900i)T^{2} \)
79 \( 1 + 0.867T + T^{2} \)
83 \( 1 + (-0.433 + 0.900i)T^{2} \)
89 \( 1 + (0.433 - 0.900i)T^{2} \)
97 \( 1 + (-0.752 - 0.752i)T + iT^{2} \)
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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

−9.658906830879874621339318760627, −8.863027308950189655447362884361, −7.84796704997939712307858185773, −7.20622775253473217040076079782, −6.38984177681419068692978765469, −5.70150013057293701274008767731, −4.39743115714272279581097148793, −3.88359935787267938791504164570, −3.09648595418760130339093857287, −1.37961015494313637916510425242, 0.891735849555700663308071448524, 2.08804186823028312676087521254, 3.02319976900179752463271548138, 4.67649634802116538574377403341, 5.49475442733277699554428230646, 5.96873693850620427194668319439, 6.64908049506561628183986414505, 7.51083098606791611344252870452, 8.371387275122401195132022640318, 9.385851745971375403911097911364

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