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Properties

Label 2-1911-1911.1217-c0-0-1
Degree $2$
Conductor $1911$
Sign $-0.380 + 0.924i$
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 + (0.781 − 0.623i)7-s + (−0.623 + 0.781i)9-s + (0.222 + 0.974i)12-s + (0.974 − 0.222i)13-s + (0.900 + 0.433i)16-s + (1.40 − 1.40i)19-s + (−0.900 − 0.433i)21-s + (−0.781 − 0.623i)25-s + (0.974 + 0.222i)27-s + (−0.900 + 0.433i)28-s + (−1.19 + 1.19i)31-s + (0.781 − 0.623i)36-s + (−1.43 − 0.900i)37-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)3-s + (−0.974 − 0.222i)4-s + (0.781 − 0.623i)7-s + (−0.623 + 0.781i)9-s + (0.222 + 0.974i)12-s + (0.974 − 0.222i)13-s + (0.900 + 0.433i)16-s + (1.40 − 1.40i)19-s + (−0.900 − 0.433i)21-s + (−0.781 − 0.623i)25-s + (0.974 + 0.222i)27-s + (−0.900 + 0.433i)28-s + (−1.19 + 1.19i)31-s + (0.781 − 0.623i)36-s + (−1.43 − 0.900i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.380 + 0.924i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.380 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8186948585\)
\(L(\frac12)\) \(\approx\) \(0.8186948585\)
\(L(1)\) 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 \( 1 + (-0.781 + 0.623i)T \)
13 \( 1 + (-0.974 + 0.222i)T \)
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 \( 1 + (1.43 + 0.900i)T + (0.433 + 0.900i)T^{2} \)
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 \( 1 + (1.05 - 0.119i)T + (0.974 - 0.222i)T^{2} \)
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

−8.819871660305706452785609292927, −8.523980422639737233963120934017, −7.42922492447909246676640013438, −7.03703165970963453892483889134, −5.69815545308865050171998160601, −5.31453267850347470318319531762, −4.34467773914555585767851777505, −3.32324295118435814363800209268, −1.76040280456931299780720413048, −0.74299796399130634368187969115, 1.48138594829664643763229513043, 3.28730106948205461488926352383, 3.89072275843933825525918379127, 4.80403518607329894013270659053, 5.56864789294672993626236638277, 6.00969076455145584219750143995, 7.57300662568625055120591462784, 8.229253614276529904725949457443, 9.008180771261057626897248265737, 9.538585238616148053780282245280

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