Properties

Label 1-23e2-529.357-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.936 - 0.350i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (0.154 − 0.987i)2-s + (−0.999 − 0.0124i)3-s + (−0.952 − 0.305i)4-s + (−0.556 + 0.831i)5-s + (−0.166 + 0.985i)6-s + (−0.635 + 0.771i)7-s + (−0.449 + 0.893i)8-s + (0.999 + 0.0248i)9-s + (0.735 + 0.678i)10-s + (−0.806 + 0.591i)11-s + (0.948 + 0.317i)12-s + (−0.514 + 0.857i)13-s + (0.664 + 0.747i)14-s + (0.566 − 0.824i)15-s + (0.813 + 0.581i)16-s + (0.984 − 0.172i)17-s + ⋯
L(s)  = 1  + (0.154 − 0.987i)2-s + (−0.999 − 0.0124i)3-s + (−0.952 − 0.305i)4-s + (−0.556 + 0.831i)5-s + (−0.166 + 0.985i)6-s + (−0.635 + 0.771i)7-s + (−0.449 + 0.893i)8-s + (0.999 + 0.0248i)9-s + (0.735 + 0.678i)10-s + (−0.806 + 0.591i)11-s + (0.948 + 0.317i)12-s + (−0.514 + 0.857i)13-s + (0.664 + 0.747i)14-s + (0.566 − 0.824i)15-s + (0.813 + 0.581i)16-s + (0.984 − 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.936 - 0.350i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.936 - 0.350i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02250664540 - 0.1244095705i\)
\(L(\frac12)\) \(\approx\) \(0.02250664540 - 0.1244095705i\)
\(L(1)\) \(\approx\) \(0.4828026579 - 0.1130257820i\)
\(L(1)\) \(\approx\) \(0.4828026579 - 0.1130257820i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.154 - 0.987i)T \)
3 \( 1 + (-0.999 - 0.0124i)T \)
5 \( 1 + (-0.556 + 0.831i)T \)
7 \( 1 + (-0.635 + 0.771i)T \)
11 \( 1 + (-0.806 + 0.591i)T \)
13 \( 1 + (-0.514 + 0.857i)T \)
17 \( 1 + (0.984 - 0.172i)T \)
19 \( 1 + (-0.834 - 0.551i)T \)
29 \( 1 + (-0.873 - 0.487i)T \)
31 \( 1 + (0.323 - 0.946i)T \)
37 \( 1 + (-0.0929 + 0.995i)T \)
41 \( 1 + (0.227 + 0.973i)T \)
43 \( 1 + (-0.215 + 0.976i)T \)
47 \( 1 + (-0.576 - 0.816i)T \)
53 \( 1 + (0.735 - 0.678i)T \)
59 \( 1 + (0.130 - 0.991i)T \)
61 \( 1 + (0.767 - 0.640i)T \)
67 \( 1 + (0.975 + 0.221i)T \)
71 \( 1 + (-0.996 - 0.0868i)T \)
73 \( 1 + (-0.873 + 0.487i)T \)
79 \( 1 + (-0.977 + 0.209i)T \)
83 \( 1 + (-0.986 - 0.160i)T \)
89 \( 1 + (0.912 + 0.409i)T \)
97 \( 1 + (-0.791 - 0.611i)T \)
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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

−23.72955544235599826250546031278, −23.197237297952443251726799569514, −22.61588243285099738314062092171, −21.52265662757933383689627243275, −20.72433242399564620431959024088, −19.39732177271622182483092451159, −18.68694135404509091637045125722, −17.529481831420862378400797995156, −16.915714692543599441292804304604, −16.242416684224884170282988267016, −15.76072680122338702515230600051, −14.66738251676768921198361871530, −13.37829388910599822322929163162, −12.728761326674699750729563741158, −12.19186372120177032535457362298, −10.6676061506849106099198426342, −10.03150694108621839853186228489, −8.770910002908986010156058545016, −7.7157893191918052372878223659, −7.16158021985786744769977029353, −5.819530400975894759285725933119, −5.35826016292754056961945343375, −4.27201230912595087377976781359, −3.433186907937367937903573165319, −0.93846603579570201308603796979, 0.095117417955768816199418823397, 1.98745585934865758682626249522, 2.8990863291318953226046565074, 4.10408585227912195526313850724, 5.01105664773744871328486489241, 6.04277166331078528244345471739, 7.00810049521871441007236350073, 8.18815788980352798353600242754, 9.78054630592379921023600021380, 9.98734351365173298912837105348, 11.30728679084245794403167278496, 11.659705815448736981856575981523, 12.5951989833413932496572718301, 13.242700450254419353312934768750, 14.69776239766335640608600648807, 15.2904988005579157850452257773, 16.39062028247395910625404154317, 17.41372591634610539754531831832, 18.45018520184492551993539100819, 18.79732558523003737857009228686, 19.49468657842204988295615722584, 20.80526338404802142327306859477, 21.67983623526646909626707833179, 22.15517460446694331759808897969, 23.126067236015850856231162066108

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