Properties

Degree $1$
Conductor $103$
Sign $-0.215 - 0.976i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.650 − 0.759i)2-s + (0.0922 − 0.995i)3-s + (−0.153 − 0.988i)4-s + (0.992 + 0.122i)5-s + (−0.696 − 0.717i)6-s + (0.881 + 0.473i)7-s + (−0.850 − 0.526i)8-s + (−0.982 − 0.183i)9-s + (0.739 − 0.673i)10-s + (0.332 + 0.943i)11-s + (−0.998 + 0.0615i)12-s + (−0.850 + 0.526i)13-s + (0.932 − 0.361i)14-s + (0.213 − 0.976i)15-s + (−0.952 + 0.303i)16-s + (−0.696 + 0.717i)17-s + ⋯
L(s,χ)  = 1  + (0.650 − 0.759i)2-s + (0.0922 − 0.995i)3-s + (−0.153 − 0.988i)4-s + (0.992 + 0.122i)5-s + (−0.696 − 0.717i)6-s + (0.881 + 0.473i)7-s + (−0.850 − 0.526i)8-s + (−0.982 − 0.183i)9-s + (0.739 − 0.673i)10-s + (0.332 + 0.943i)11-s + (−0.998 + 0.0615i)12-s + (−0.850 + 0.526i)13-s + (0.932 − 0.361i)14-s + (0.213 − 0.976i)15-s + (−0.952 + 0.303i)16-s + (−0.696 + 0.717i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.215 - 0.976i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.215 - 0.976i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.215 - 0.976i$
Motivic weight: \(0\)
Character: $\chi_{103} (25, \cdot )$
Sato-Tate group: $\mu(51)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.215 - 0.976i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.9929917080 - 1.235558488i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.9929917080 - 1.235558488i\)
\(L(\chi,1)\) \(\approx\) \(1.225005191 - 0.9465037722i\)
\(L(1,\chi)\) \(\approx\) \(1.225005191 - 0.9465037722i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.12027477347905117949764674933, −29.343247560014340488621355119544, −27.64315917401023473511641033927, −26.87596650084128813785495652635, −25.95821279377054044955851796121, −24.85175172917922619377124166340, −24.09285446983861734670003099174, −22.592814117292056175407272489967, −21.83388022605288684570935439226, −21.05963862027546685093558482237, −20.13603833555008079521006541353, −17.97943637167788395711577040533, −17.061609689639070202757636383371, −16.3788723081014284740432244592, −14.9391964484153066254193835172, −14.244871960662246641635997393923, −13.34211192902799462932789367875, −11.6768584028346049665728517769, −10.42021714697035230021541679327, −9.06410946935062949225129235102, −8.00744403971961772272720324061, −6.27199181889312935905792793631, −5.187564228559758072197923216263, −4.2371847439787004229940648831, −2.64014859037645278115370700799, 1.88233119473528458999127937938, 2.25650891670276969814875890957, 4.49939276416839696566749872165, 5.80684936973811115927279086021, 6.87505755355599233480130359873, 8.68002939198357995533169695269, 9.9172053339943244263570523214, 11.33412841613099679032642348, 12.30996976055218791731471306305, 13.231619051729269295640358805493, 14.34426303870097565178142790564, 14.95143645126923321156176866227, 17.40316038412069414248245633174, 17.91689088913741395182099478212, 19.14851851592429417636119329591, 20.083252190444575840030858428833, 21.28923855004737826134120685649, 22.05741504061528238577888640560, 23.2927542046733562938388562136, 24.38343888189157712656078053365, 24.93927251641818915093459188143, 26.29589780063594053600233564181, 28.082132141523199257362123339948, 28.60949110737421116581254047280, 29.83080645757086298964031624334

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