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Properties

Label 1-103-103.33-r0-0-0
Degree $1$
Conductor $103$
Sign $-0.215 + 0.976i$
Analytic cond. $0.478329$
Root an. cond. $0.478329$
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  + (0.650 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.215 + 0.976i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.650 + 0.759i)T \)
3 \( 1 + (0.0922 + 0.995i)T \)
5 \( 1 + (0.992 - 0.122i)T \)
7 \( 1 + (0.881 - 0.473i)T \)
11 \( 1 + (0.332 - 0.943i)T \)
13 \( 1 + (-0.850 - 0.526i)T \)
17 \( 1 + (-0.696 - 0.717i)T \)
19 \( 1 + (-0.908 + 0.417i)T \)
23 \( 1 + (-0.982 - 0.183i)T \)
29 \( 1 + (-0.389 + 0.920i)T \)
31 \( 1 + (0.739 - 0.673i)T \)
37 \( 1 + (0.445 + 0.895i)T \)
41 \( 1 + (0.992 + 0.122i)T \)
43 \( 1 + (-0.998 + 0.0615i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.816 + 0.577i)T \)
59 \( 1 + (0.881 + 0.473i)T \)
61 \( 1 + (-0.273 + 0.961i)T \)
67 \( 1 + (-0.0307 - 0.999i)T \)
71 \( 1 + (-0.389 - 0.920i)T \)
73 \( 1 + (-0.602 + 0.798i)T \)
79 \( 1 + (-0.602 - 0.798i)T \)
83 \( 1 + (-0.0307 + 0.999i)T \)
89 \( 1 + (0.932 + 0.361i)T \)
97 \( 1 + (-0.696 + 0.717i)T \)
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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

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

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