↑

Properties

Label 1-3503-3503.174-r0-0-0
Degree $1$
Conductor $3503$
Sign $-0.706 - 0.707i$
Analytic cond. $16.2678$
Root an. cond. $16.2678$
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  + (0.691 + 0.722i)2-s + (0.961 + 0.273i)3-s + (−0.0448 + 0.998i)4-s + (−0.0373 + 0.999i)5-s + (0.467 + 0.884i)6-s + (0.992 + 0.119i)7-s + (−0.753 + 0.657i)8-s + (0.850 + 0.525i)9-s + (−0.748 + 0.663i)10-s + (−0.967 + 0.251i)11-s + (−0.316 + 0.948i)12-s + (−0.999 − 0.0149i)13-s + (0.599 + 0.800i)14-s + (−0.309 + 0.951i)15-s + (−0.995 − 0.0896i)16-s + (−0.996 + 0.0821i)17-s + ⋯
L(s)  = 1  + (0.691 + 0.722i)2-s + (0.961 + 0.273i)3-s + (−0.0448 + 0.998i)4-s + (−0.0373 + 0.999i)5-s + (0.467 + 0.884i)6-s + (0.992 + 0.119i)7-s + (−0.753 + 0.657i)8-s + (0.850 + 0.525i)9-s + (−0.748 + 0.663i)10-s + (−0.967 + 0.251i)11-s + (−0.316 + 0.948i)12-s + (−0.999 − 0.0149i)13-s + (0.599 + 0.800i)14-s + (−0.309 + 0.951i)15-s + (−0.995 − 0.0896i)16-s + (−0.996 + 0.0821i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3503 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3503\)    =    \(31 \cdot 113\)
Sign: $-0.706 - 0.707i$
Analytic conductor: \(16.2678\)
Root analytic conductor: \(16.2678\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3503} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3503,\ (0:\ ),\ -0.706 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9354328370 + 2.256416765i\)
\(L(\frac12)\) \(\approx\) \(-0.9354328370 + 2.256416765i\)
\(L(1)\) \(\approx\) \(1.077792787 + 1.401044170i\)
\(L(1)\) \(\approx\) \(1.077792787 + 1.401044170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
113 \( 1 \)
good2 \( 1 + (0.691 + 0.722i)T \)
3 \( 1 + (0.961 + 0.273i)T \)
5 \( 1 + (-0.0373 + 0.999i)T \)
7 \( 1 + (0.992 + 0.119i)T \)
11 \( 1 + (-0.967 + 0.251i)T \)
13 \( 1 + (-0.999 - 0.0149i)T \)
17 \( 1 + (-0.996 + 0.0821i)T \)
19 \( 1 + (-0.344 - 0.938i)T \)
23 \( 1 + (0.928 - 0.372i)T \)
29 \( 1 + (-0.0224 + 0.999i)T \)
37 \( 1 + (-0.593 + 0.804i)T \)
41 \( 1 + (-0.986 + 0.163i)T \)
43 \( 1 + (-0.519 + 0.854i)T \)
47 \( 1 + (0.979 + 0.200i)T \)
53 \( 1 + (-0.712 - 0.701i)T \)
59 \( 1 + (0.273 - 0.961i)T \)
61 \( 1 + (0.781 - 0.623i)T \)
67 \( 1 + (0.916 - 0.399i)T \)
71 \( 1 + (-0.777 + 0.629i)T \)
73 \( 1 + (-0.933 + 0.358i)T \)
79 \( 1 + (-0.862 + 0.506i)T \)
83 \( 1 + (-0.280 + 0.959i)T \)
89 \( 1 + (0.822 + 0.569i)T \)
97 \( 1 + (0.753 + 0.657i)T \)
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

−18.66879004146420596299646284467, −17.70773040161819830103937112496, −17.10495987533683460373571944353, −15.91689778373470872692654551741, −15.3291637900901127232789973176, −14.76792571014149741293175347534, −13.985677162793889756003943382178, −13.38041397335420651235937845178, −12.936335251112897203125873879242, −12.12777080050259544021992003204, −11.62768701402109153701403635368, −10.53915672963589216419761274463, −10.0460502259573640157194197820, −9.007683482209189664376212714711, −8.64059283203710022920317819120, −7.72198399917961712007844310485, −7.09958363785753453601873780966, −5.81577909723910320489693446563, −5.1180584953987008133730114497, −4.468394844998578909628414281970, −3.87429262406109442268339863167, −2.78605723913404572292250528187, −2.07627145719382545890077695358, −1.53903139495880124340277781482, −0.40788763915380205496540090776, 1.92557370672014294804352671573, 2.59003716622311598650555563958, 3.0473631206312876751679643465, 4.09985692990424529176298108503, 4.92028846137126960276024758312, 5.1666513473107588815656810756, 6.76333037983294648112761662097, 6.955296162393922528184480285004, 7.83010880399684406349028913979, 8.35951192882732283470681856585, 9.10275463199287370258753570010, 10.08381860517732017597560852899, 10.89544726318346054685443031558, 11.436141566581428675911321379643, 12.556253706490186159241767283031, 13.1731483568840865000706277488, 13.84735052302851124952543791951, 14.56335963164303741015856805636, 14.963612964986220210915877583616, 15.41377714409825058095293111417, 16.042782298581438892506235545744, 17.24662998445733421116741885759, 17.64740024686583778953208394933, 18.4739065424026173167042973162, 19.05047454137711599166700474035

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