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Properties

Label 1-919-919.58-r0-0-0
Degree $1$
Conductor $919$
Sign $0.422 + 0.906i$
Analytic cond. $4.26781$
Root an. cond. $4.26781$
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  + (−0.273 + 0.961i)2-s + (0.739 − 0.673i)3-s + (−0.850 − 0.526i)4-s + (0.739 − 0.673i)5-s + (0.445 + 0.895i)6-s + (0.445 + 0.895i)7-s + (0.739 − 0.673i)8-s + (0.0922 − 0.995i)9-s + (0.445 + 0.895i)10-s + (0.0922 + 0.995i)11-s + (−0.982 + 0.183i)12-s + (−0.850 + 0.526i)13-s + (−0.982 + 0.183i)14-s + (0.0922 − 0.995i)15-s + (0.445 + 0.895i)16-s + (0.932 + 0.361i)17-s + ⋯
L(s)  = 1  + (−0.273 + 0.961i)2-s + (0.739 − 0.673i)3-s + (−0.850 − 0.526i)4-s + (0.739 − 0.673i)5-s + (0.445 + 0.895i)6-s + (0.445 + 0.895i)7-s + (0.739 − 0.673i)8-s + (0.0922 − 0.995i)9-s + (0.445 + 0.895i)10-s + (0.0922 + 0.995i)11-s + (−0.982 + 0.183i)12-s + (−0.850 + 0.526i)13-s + (−0.982 + 0.183i)14-s + (0.0922 − 0.995i)15-s + (0.445 + 0.895i)16-s + (0.932 + 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(919\)
Sign: $0.422 + 0.906i$
Analytic conductor: \(4.26781\)
Root analytic conductor: \(4.26781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{919} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 919,\ (0:\ ),\ 0.422 + 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.543518534 + 0.9833289334i\)
\(L(\frac12)\) \(\approx\) \(1.543518534 + 0.9833289334i\)
\(L(1)\) \(\approx\) \(1.235620028 + 0.4128690374i\)
\(L(1)\) \(\approx\) \(1.235620028 + 0.4128690374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad919 \( 1 \)
good2 \( 1 + (-0.273 + 0.961i)T \)
3 \( 1 + (0.739 - 0.673i)T \)
5 \( 1 + (0.739 - 0.673i)T \)
7 \( 1 + (0.445 + 0.895i)T \)
11 \( 1 + (0.0922 + 0.995i)T \)
13 \( 1 + (-0.850 + 0.526i)T \)
17 \( 1 + (0.932 + 0.361i)T \)
19 \( 1 + (0.739 + 0.673i)T \)
23 \( 1 + (0.0922 + 0.995i)T \)
29 \( 1 + (-0.273 + 0.961i)T \)
31 \( 1 + (0.445 + 0.895i)T \)
37 \( 1 + (-0.850 + 0.526i)T \)
41 \( 1 + (-0.982 - 0.183i)T \)
43 \( 1 + (0.739 + 0.673i)T \)
47 \( 1 + (-0.850 - 0.526i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.850 - 0.526i)T \)
61 \( 1 + (0.445 - 0.895i)T \)
67 \( 1 + (-0.602 - 0.798i)T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (-0.602 - 0.798i)T \)
79 \( 1 + (0.445 - 0.895i)T \)
83 \( 1 + (0.932 - 0.361i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (0.0922 - 0.995i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.48188263970314579958192329407, −20.91054554788112376635539968291, −20.34144762229970067824379058283, −19.40557862848853189004341581021, −18.85144903484892026282453037489, −17.89482097301691303404382924097, −17.04892866943771206723840763529, −16.483600796446673325546097496790, −15.08992588872732500652168255326, −14.25706171289340281429871681727, −13.7896232143739374139181391143, −13.16930060778173678485814452632, −11.78964348500401294846841494972, −10.95235796027534754072248345062, −10.263938199934629300322839401776, −9.80428599902716360914313587643, −8.88510464427972170050229384081, −7.91078374353066716777248824287, −7.23927019766788165831161746022, −5.588402842586973818129511915173, −4.73769961911036493225052854841, −3.683353663978662595109961778282, −2.943190014886703345041281708395, −2.22252019102103799044938118848, −0.86475718121769811892409335849, 1.454639833055684496017373791505, 1.84217592660176760808371577991, 3.36346634189719999353511020914, 4.816003911283798719636315309376, 5.37851618020518793650563405180, 6.38645551552570271248386724016, 7.30242366925608670349237018125, 8.02970284826112898131338289693, 8.86408857967096568237052265928, 9.50977067537557703120637396119, 10.09199331174610996010638263567, 12.07278484968285769917372438344, 12.39430223141937413635302902380, 13.42413873241172922911683676220, 14.264109553511306227081236277616, 14.70442843374277169850462133163, 15.54625383538933419393437019942, 16.56443651126024348910919696548, 17.4538582474399602176918515634, 17.92521016852388466704426239599, 18.708068927958029617792834926364, 19.47575258662660431431223769399, 20.359407432505159004580901360166, 21.2582268178343141622519904051, 21.97864168291223973845100790682

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