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Properties

Label 2-1911-1911.1328-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.992 + 0.121i$
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  + (−0.997 − 0.0747i)3-s + (0.974 + 0.222i)4-s + (−0.149 − 0.988i)7-s + (0.988 + 0.149i)9-s + (−0.955 − 0.294i)12-s + (0.680 + 0.733i)13-s + (0.900 + 0.433i)16-s + (−0.416 + 1.55i)19-s + (0.0747 + 0.997i)21-s + (0.149 − 0.988i)25-s + (−0.974 − 0.222i)27-s + (0.0747 − 0.997i)28-s + (0.275 − 1.02i)31-s + (0.930 + 0.365i)36-s + (1.06 − 1.69i)37-s + ⋯
L(s)  = 1  + (−0.997 − 0.0747i)3-s + (0.974 + 0.222i)4-s + (−0.149 − 0.988i)7-s + (0.988 + 0.149i)9-s + (−0.955 − 0.294i)12-s + (0.680 + 0.733i)13-s + (0.900 + 0.433i)16-s + (−0.416 + 1.55i)19-s + (0.0747 + 0.997i)21-s + (0.149 − 0.988i)25-s + (−0.974 − 0.222i)27-s + (0.0747 − 0.997i)28-s + (0.275 − 1.02i)31-s + (0.930 + 0.365i)36-s + (1.06 − 1.69i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.992 + 0.121i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.992 + 0.121i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103345909\)
\(L(\frac12)\) \(\approx\) \(1.103345909\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.997 + 0.0747i)T \)
7 \( 1 + (0.149 + 0.988i)T \)
13 \( 1 + (-0.680 - 0.733i)T \)
good2 \( 1 + (-0.974 - 0.222i)T^{2} \)
5 \( 1 + (-0.149 + 0.988i)T^{2} \)
11 \( 1 + (-0.294 - 0.955i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 + (0.416 - 1.55i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (-0.0747 + 0.997i)T^{2} \)
31 \( 1 + (-0.275 + 1.02i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (-1.06 + 1.69i)T + (-0.433 - 0.900i)T^{2} \)
41 \( 1 + (0.930 - 0.365i)T^{2} \)
43 \( 1 + (-0.167 - 0.246i)T + (-0.365 + 0.930i)T^{2} \)
47 \( 1 + (0.294 + 0.955i)T^{2} \)
53 \( 1 + (-0.826 + 0.563i)T^{2} \)
59 \( 1 + (-0.781 + 0.623i)T^{2} \)
61 \( 1 + (-1.29 - 1.40i)T + (-0.0747 + 0.997i)T^{2} \)
67 \( 1 + (-0.170 - 0.638i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.997 + 0.0747i)T^{2} \)
73 \( 1 + (-1.42 + 1.05i)T + (0.294 - 0.955i)T^{2} \)
79 \( 1 + (0.974 - 1.68i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.974 + 0.222i)T^{2} \)
89 \( 1 + (0.974 - 0.222i)T^{2} \)
97 \( 1 + (1.14 - 0.307i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.747986862857657681502052737602, −8.353693454750681699659831956952, −7.61800738613571470904593465574, −6.92083196476816842070277729170, −6.22562734014135202720771697884, −5.70982586969283451892345187977, −4.22692404957499471327081804915, −3.85392090382750710690152985815, −2.27514150414405955266854474229, −1.17021787070133487296656500028, 1.20681533098195492385747929647, 2.48185267536342397296272519749, 3.41956434490280964819785740201, 4.90655266686717240995941121681, 5.42355897933584178010706111454, 6.37646653318129132933987393443, 6.67842139134645024585167295634, 7.71727044761630248471721412084, 8.652437382260965852451150885173, 9.565948236601438136523414646563

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