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Properties

Label 2-1911-1911.1025-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.715 + 0.698i$
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  + (0.974 + 0.222i)3-s + (0.149 − 0.988i)4-s + (0.0747 − 0.997i)7-s + (0.900 + 0.433i)9-s + (0.365 − 0.930i)12-s + (0.680 + 0.733i)13-s + (−0.955 − 0.294i)16-s + (−0.262 + 0.262i)19-s + (0.294 − 0.955i)21-s + (−0.563 − 0.826i)25-s + (0.781 + 0.623i)27-s + (−0.974 − 0.222i)28-s + (0.0579 + 0.216i)31-s + (0.563 − 0.826i)36-s + (−0.955 + 1.29i)37-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)3-s + (0.149 − 0.988i)4-s + (0.0747 − 0.997i)7-s + (0.900 + 0.433i)9-s + (0.365 − 0.930i)12-s + (0.680 + 0.733i)13-s + (−0.955 − 0.294i)16-s + (−0.262 + 0.262i)19-s + (0.294 − 0.955i)21-s + (−0.563 − 0.826i)25-s + (0.781 + 0.623i)27-s + (−0.974 − 0.222i)28-s + (0.0579 + 0.216i)31-s + (0.563 − 0.826i)36-s + (−0.955 + 1.29i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.715 + 0.698i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.715 + 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.698738382\)
\(L(\frac12)\) \(\approx\) \(1.698738382\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.974 - 0.222i)T \)
7 \( 1 + (-0.0747 + 0.997i)T \)
13 \( 1 + (-0.680 - 0.733i)T \)
good2 \( 1 + (-0.149 + 0.988i)T^{2} \)
5 \( 1 + (0.563 + 0.826i)T^{2} \)
11 \( 1 + (0.781 + 0.623i)T^{2} \)
17 \( 1 + (0.733 + 0.680i)T^{2} \)
19 \( 1 + (0.262 - 0.262i)T - iT^{2} \)
23 \( 1 + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.733 + 0.680i)T^{2} \)
31 \( 1 + (-0.0579 - 0.216i)T + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (0.955 - 1.29i)T + (-0.294 - 0.955i)T^{2} \)
41 \( 1 + (0.563 + 0.826i)T^{2} \)
43 \( 1 + (-0.332 + 1.07i)T + (-0.826 - 0.563i)T^{2} \)
47 \( 1 + (0.930 - 0.365i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.997 - 0.0747i)T^{2} \)
61 \( 1 + (-0.571 + 0.455i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (0.752 + 0.752i)T + iT^{2} \)
71 \( 1 + (0.680 + 0.733i)T^{2} \)
73 \( 1 + (0.785 + 0.148i)T + (0.930 + 0.365i)T^{2} \)
79 \( 1 + (-0.781 - 1.35i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.781 + 0.623i)T^{2} \)
89 \( 1 + (-0.930 - 0.365i)T^{2} \)
97 \( 1 + (-0.508 - 1.89i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.325860175477327730871688888002, −8.612777053770182216501785513845, −7.80514265028626642966669085888, −6.88466411733372854706415214191, −6.33498089305548668228567238160, −5.09365999655653371949067952304, −4.28591682469329518045474374972, −3.54965990412060571918652396598, −2.21378721476558440058540459167, −1.30307578909737407460527533323, 1.81002978347308568977483852859, 2.75608201681361316182647147088, 3.43273913515676503716361087193, 4.32095404567633683722880262879, 5.54917240515022003622139662718, 6.46810423151261150929650599519, 7.44215499462213612868896604707, 7.940829360048169572910334698708, 8.816313565754780105706286123161, 9.022683118897887727188175511949

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