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Properties

Label 2-1911-1911.629-c0-0-1
Degree $2$
Conductor $1911$
Sign $0.889 + 0.457i$
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  + (−0.433 − 0.900i)3-s + (0.974 + 0.222i)4-s + (0.623 + 0.781i)7-s + (−0.623 + 0.781i)9-s + (−0.222 − 0.974i)12-s + (0.433 − 0.900i)13-s + (0.900 + 0.433i)16-s + (−1.40 − 1.40i)19-s + (0.433 − 0.900i)21-s + (0.781 + 0.623i)25-s + (0.974 + 0.222i)27-s + (0.433 + 0.900i)28-s + (1.19 + 1.19i)31-s + (−0.781 + 0.623i)36-s + (0.900 − 1.43i)37-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)3-s + (0.974 + 0.222i)4-s + (0.623 + 0.781i)7-s + (−0.623 + 0.781i)9-s + (−0.222 − 0.974i)12-s + (0.433 − 0.900i)13-s + (0.900 + 0.433i)16-s + (−1.40 − 1.40i)19-s + (0.433 − 0.900i)21-s + (0.781 + 0.623i)25-s + (0.974 + 0.222i)27-s + (0.433 + 0.900i)28-s + (1.19 + 1.19i)31-s + (−0.781 + 0.623i)36-s + (0.900 − 1.43i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.889 + 0.457i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.889 + 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.342627031\)
\(L(\frac12)\) \(\approx\) \(1.342627031\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.433 + 0.900i)T \)
7 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (-0.433 + 0.900i)T \)
good2 \( 1 + (-0.974 - 0.222i)T^{2} \)
5 \( 1 + (-0.781 - 0.623i)T^{2} \)
11 \( 1 + (0.974 + 0.222i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 + (1.40 + 1.40i)T + iT^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (-1.19 - 1.19i)T + iT^{2} \)
37 \( 1 + (-0.900 + 1.43i)T + (-0.433 - 0.900i)T^{2} \)
41 \( 1 + (-0.781 - 0.623i)T^{2} \)
43 \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.974 - 0.222i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.781 + 0.623i)T^{2} \)
61 \( 1 + (0.433 - 0.0990i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-1.33 + 1.33i)T - iT^{2} \)
71 \( 1 + (0.433 - 0.900i)T^{2} \)
73 \( 1 + (0.119 + 1.05i)T + (-0.974 + 0.222i)T^{2} \)
79 \( 1 + 1.94T + T^{2} \)
83 \( 1 + (-0.974 + 0.222i)T^{2} \)
89 \( 1 + (0.974 - 0.222i)T^{2} \)
97 \( 1 + (0.158 + 0.158i)T + iT^{2} \)
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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

−9.069461020759765572372245277468, −8.316714233475064938423614709278, −7.80873218189820654173280177863, −6.83920222662936849447471256004, −6.32866234294700567987733171649, −5.50992780352555602462229344153, −4.67165921662930019052676881143, −3.01176110627204262949851672611, −2.39572908894048676199247396995, −1.29740771872686422092232303469, 1.34729633229788680271414708072, 2.58423396936077715589664848412, 3.91004007574990409902234762171, 4.38715563511724473984896445405, 5.48609360755794780059129163851, 6.38038409846357194633931114598, 6.78689222355352354865459965242, 8.046395541746476667566889293607, 8.545206153869274089426487517951, 9.823818765136839813808699820228

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