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Properties

Label 2-1911-1911.1388-c0-0-0
Degree $2$
Conductor $1911$
Sign $-0.996 + 0.0832i$
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  + (−0.826 + 0.563i)3-s + (0.222 + 0.974i)4-s + (−0.365 − 0.930i)7-s + (0.365 − 0.930i)9-s + (−0.733 − 0.680i)12-s + (−0.955 + 0.294i)13-s + (−0.900 + 0.433i)16-s + (−1.61 + 0.930i)19-s + (0.826 + 0.563i)21-s + (−0.365 + 0.930i)25-s + (0.222 + 0.974i)27-s + (0.826 − 0.563i)28-s + (−1.68 + 0.974i)31-s + (0.988 + 0.149i)36-s + (−0.574 − 0.131i)37-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)3-s + (0.222 + 0.974i)4-s + (−0.365 − 0.930i)7-s + (0.365 − 0.930i)9-s + (−0.733 − 0.680i)12-s + (−0.955 + 0.294i)13-s + (−0.900 + 0.433i)16-s + (−1.61 + 0.930i)19-s + (0.826 + 0.563i)21-s + (−0.365 + 0.930i)25-s + (0.222 + 0.974i)27-s + (0.826 − 0.563i)28-s + (−1.68 + 0.974i)31-s + (0.988 + 0.149i)36-s + (−0.574 − 0.131i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.996 + 0.0832i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.996 + 0.0832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3022206405\)
\(L(\frac12)\) \(\approx\) \(0.3022206405\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.826 - 0.563i)T \)
7 \( 1 + (0.365 + 0.930i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
good2 \( 1 + (-0.222 - 0.974i)T^{2} \)
5 \( 1 + (0.365 - 0.930i)T^{2} \)
11 \( 1 + (-0.733 + 0.680i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 + (1.61 - 0.930i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
29 \( 1 + (-0.826 + 0.563i)T^{2} \)
31 \( 1 + (1.68 - 0.974i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.574 + 0.131i)T + (0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.988 + 0.149i)T^{2} \)
43 \( 1 + (-0.0546 + 0.728i)T + (-0.988 - 0.149i)T^{2} \)
47 \( 1 + (-0.733 + 0.680i)T^{2} \)
53 \( 1 + (-0.0747 + 0.997i)T^{2} \)
59 \( 1 + (0.623 - 0.781i)T^{2} \)
61 \( 1 + (-1.40 + 0.432i)T + (0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.751 + 0.433i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.826 + 0.563i)T^{2} \)
73 \( 1 + (-1.26 - 0.496i)T + (0.733 + 0.680i)T^{2} \)
79 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.222 + 0.974i)T^{2} \)
97 \( 1 + (1.61 + 0.930i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.868491761801804551805752857310, −9.113589851447734527692796682975, −8.209359924957898638365322947693, −7.13007787092336573790078391066, −6.91301803713493634495784677547, −5.82523679185575925424482308929, −4.80763193896657876368743044199, −3.94234568530735494373736669865, −3.46206878925705316032644466071, −1.93966281416635327447204929673, 0.22303493643666670201241049354, 1.96049250994294638512533579447, 2.53677299737415282489767288347, 4.35902571409487841572701827836, 5.20249843576936269125482955829, 5.82529546789004751298330759944, 6.51548389867003141924266381898, 7.13492367697901246619591040137, 8.182853234386473037459195913477, 9.122647346180497694706407396945

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