↑

Properties

Label 1-103-103.28-r0-0-0
Degree $1$
Conductor $103$
Sign $0.874 - 0.485i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.389 − 0.920i)2-s + (0.445 + 0.895i)3-s + (−0.696 + 0.717i)4-s + (0.816 − 0.577i)5-s + (0.650 − 0.759i)6-s + (−0.779 − 0.626i)7-s + (0.932 + 0.361i)8-s + (−0.602 + 0.798i)9-s + (−0.850 − 0.526i)10-s + (0.992 + 0.122i)11-s + (−0.952 − 0.303i)12-s + (0.932 − 0.361i)13-s + (−0.273 + 0.961i)14-s + (0.881 + 0.473i)15-s + (−0.0307 − 0.999i)16-s + (0.650 + 0.759i)17-s + ⋯
L(s)  = 1  + (−0.389 − 0.920i)2-s + (0.445 + 0.895i)3-s + (−0.696 + 0.717i)4-s + (0.816 − 0.577i)5-s + (0.650 − 0.759i)6-s + (−0.779 − 0.626i)7-s + (0.932 + 0.361i)8-s + (−0.602 + 0.798i)9-s + (−0.850 − 0.526i)10-s + (0.992 + 0.122i)11-s + (−0.952 − 0.303i)12-s + (0.932 − 0.361i)13-s + (−0.273 + 0.961i)14-s + (0.881 + 0.473i)15-s + (−0.0307 − 0.999i)16-s + (0.650 + 0.759i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.874 - 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9759537071 - 0.2527038303i\)
\(L(\frac12)\) \(\approx\) \(0.9759537071 - 0.2527038303i\)
\(L(1)\) \(\approx\) \(0.9990465318 - 0.2141575636i\)
\(L(1)\) \(\approx\) \(0.9990465318 - 0.2141575636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.389 - 0.920i)T \)
3 \( 1 + (0.445 + 0.895i)T \)
5 \( 1 + (0.816 - 0.577i)T \)
7 \( 1 + (-0.779 - 0.626i)T \)
11 \( 1 + (0.992 + 0.122i)T \)
13 \( 1 + (0.932 - 0.361i)T \)
17 \( 1 + (0.650 + 0.759i)T \)
19 \( 1 + (0.552 + 0.833i)T \)
23 \( 1 + (-0.602 - 0.798i)T \)
29 \( 1 + (-0.908 - 0.417i)T \)
31 \( 1 + (-0.850 + 0.526i)T \)
37 \( 1 + (0.739 - 0.673i)T \)
41 \( 1 + (0.816 + 0.577i)T \)
43 \( 1 + (-0.952 + 0.303i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.998 + 0.0615i)T \)
59 \( 1 + (-0.779 + 0.626i)T \)
61 \( 1 + (-0.982 + 0.183i)T \)
67 \( 1 + (-0.153 - 0.988i)T \)
71 \( 1 + (-0.908 + 0.417i)T \)
73 \( 1 + (0.0922 - 0.995i)T \)
79 \( 1 + (0.0922 + 0.995i)T \)
83 \( 1 + (-0.153 + 0.988i)T \)
89 \( 1 + (-0.273 + 0.961i)T \)
97 \( 1 + (0.650 - 0.759i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.81524478313759959950806641328, −28.89664953484593681698911256056, −27.78519849543662769364615817523, −26.25780025027430783025331724300, −25.62887339324447858641839271631, −25.04534497523904440518537582828, −23.98070538994780898728443517387, −22.7871663505293540528455866028, −21.91086201714676116855398533675, −20.10725917677385247199780973048, −18.888656571008812493251102497883, −18.3992266165329414073152775003, −17.361142906311168564624365611884, −16.11438119178031680451143491271, −14.79600038231929230854434426982, −13.90628216259961747253712484446, −13.11488924119541630175318749132, −11.45299664779597736865019150064, −9.5188410571428327780224734362, −9.07776650931613272958445188763, −7.460097028924524999472335710004, −6.45715336868776417045638279511, −5.74400638455585415676763811541, −3.31201691537793491581744198207, −1.56464630265273853212345963929, 1.55382266777160486578811397669, 3.33608708201712914588594674801, 4.21420125963893212834433181383, 5.89912810708144480590377291165, 8.065565043360590853651071912, 9.2271693854615983221721766713, 9.91021452636347298549558863463, 10.84317964122401173352678534764, 12.45383304296877962066702916734, 13.502865331881647075126977433, 14.446096458280805394945094444233, 16.45104057597644420212059710946, 16.77324605923234936933948547690, 18.219570454579059737568828685172, 19.65738615319283374068035743970, 20.30815377790117681601933898657, 21.16097327357818035138726077302, 22.140041138752673617411478054494, 23.0176061978908638422319540005, 25.06077419435796233727434872724, 25.83671033950593259030617284535, 26.70889844855407744439664096924, 27.903423344494411562734922187692, 28.47016931387097475261976599683, 29.69181462548515247529009992690

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