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Properties

Label 1-1903-1903.989-r1-0-0
Degree $1$
Conductor $1903$
Sign $-0.390 + 0.920i$
Analytic cond. $204.505$
Root an. cond. $204.505$
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  + (0.791 + 0.611i)2-s + (−0.457 + 0.889i)3-s + (0.252 + 0.967i)4-s + (−0.872 − 0.489i)5-s + (−0.905 + 0.424i)6-s + (0.934 − 0.357i)7-s + (−0.391 + 0.920i)8-s + (−0.581 − 0.813i)9-s + (−0.391 − 0.920i)10-s + (−0.976 − 0.217i)12-s + (0.791 + 0.611i)13-s + (0.957 + 0.288i)14-s + (0.833 − 0.551i)15-s + (−0.872 + 0.489i)16-s + (−0.639 − 0.768i)17-s + (0.0365 − 0.999i)18-s + ⋯
L(s)  = 1  + (0.791 + 0.611i)2-s + (−0.457 + 0.889i)3-s + (0.252 + 0.967i)4-s + (−0.872 − 0.489i)5-s + (−0.905 + 0.424i)6-s + (0.934 − 0.357i)7-s + (−0.391 + 0.920i)8-s + (−0.581 − 0.813i)9-s + (−0.391 − 0.920i)10-s + (−0.976 − 0.217i)12-s + (0.791 + 0.611i)13-s + (0.957 + 0.288i)14-s + (0.833 − 0.551i)15-s + (−0.872 + 0.489i)16-s + (−0.639 − 0.768i)17-s + (0.0365 − 0.999i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1903 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1903 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1903\)    =    \(11 \cdot 173\)
Sign: $-0.390 + 0.920i$
Analytic conductor: \(204.505\)
Root analytic conductor: \(204.505\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1903} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1903,\ (1:\ ),\ -0.390 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.348494883 + 2.036130847i\)
\(L(\frac12)\) \(\approx\) \(1.348494883 + 2.036130847i\)
\(L(1)\) \(\approx\) \(1.074071060 + 0.7306455763i\)
\(L(1)\) \(\approx\) \(1.074071060 + 0.7306455763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
173 \( 1 \)
good2 \( 1 + (0.791 + 0.611i)T \)
3 \( 1 + (-0.457 + 0.889i)T \)
5 \( 1 + (-0.872 - 0.489i)T \)
7 \( 1 + (0.934 - 0.357i)T \)
13 \( 1 + (0.791 + 0.611i)T \)
17 \( 1 + (-0.639 - 0.768i)T \)
19 \( 1 + (-0.957 + 0.288i)T \)
23 \( 1 + (0.833 + 0.551i)T \)
29 \( 1 + (-0.905 - 0.424i)T \)
31 \( 1 + (-0.457 - 0.889i)T \)
37 \( 1 + (0.957 - 0.288i)T \)
41 \( 1 + (0.934 - 0.357i)T \)
43 \( 1 + (-0.252 + 0.967i)T \)
47 \( 1 + (-0.322 - 0.946i)T \)
53 \( 1 + (-0.181 - 0.983i)T \)
59 \( 1 + (0.744 - 0.667i)T \)
61 \( 1 + (-0.639 + 0.768i)T \)
67 \( 1 + (-0.457 + 0.889i)T \)
71 \( 1 + (0.905 + 0.424i)T \)
73 \( 1 + (0.694 - 0.719i)T \)
79 \( 1 + (0.322 - 0.946i)T \)
83 \( 1 + (-0.520 + 0.853i)T \)
89 \( 1 + (0.989 - 0.145i)T \)
97 \( 1 + (0.520 + 0.853i)T \)
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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

−19.74827318602123787916438577532, −18.86859924195543358240252320958, −18.443834631103578684562429722427, −17.75713966997002697910997596361, −16.76915344800504787212914768651, −15.706794711482708469124043898845, −15.01598856663789136144170511677, −14.52565082513992803513783778199, −13.59801670570269378801378616520, −12.7123325761784932897762848705, −12.434986659140300601965659866666, −11.34367924995174033980974005538, −10.949756085462800430525020582759, −10.64793543647455046543375031059, −8.938182856207940860585424271142, −8.24666614750244071898819948234, −7.36868944768401903647000645338, −6.51863911324016641224595026207, −5.88499159107061610809730732885, −4.92421948190247478445926273444, −4.21892241170916938353349387371, −3.16413249011498240838093227813, −2.31889928758922585116580811947, −1.46751680614064941148181316005, −0.52365677561840696147080854456, 0.65908071288685101941630772431, 2.13114808101531185079932412550, 3.51075819392063120068041000719, 4.09150130847134135495099836374, 4.62436127183257734890210177567, 5.323026124466014160496308806387, 6.21972343913382144574986208818, 7.15609291344035866987687449020, 7.956890172487571204885289083111, 8.7305429671788218475860404686, 9.369338600066080576629108456238, 10.871520165018721017956343609982, 11.380146098859097268195338548032, 11.668101486954102418936867633525, 12.84204413096632457864109262941, 13.464507228136319215463536554869, 14.59634371011689514742564286553, 14.94437617043795111961987693246, 15.70440160928685965335744460585, 16.41534358279782820190757932007, 16.830182172794837137552299259723, 17.605440843586975482794057973507, 18.43628998673043181211775237609, 19.66823668961492329380421082145, 20.52675763067513481341711563827

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