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Properties

Label 1-4013-4013.831-r0-0-0
Degree $1$
Conductor $4013$
Sign $-0.0761 + 0.997i$
Analytic cond. $18.6362$
Root an. cond. $18.6362$
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  + (0.602 + 0.798i)2-s + (−0.739 − 0.673i)3-s + (−0.273 + 0.961i)4-s + (−0.932 − 0.361i)5-s + (0.0922 − 0.995i)6-s + (0.739 + 0.673i)7-s + (−0.932 + 0.361i)8-s + (0.0922 + 0.995i)9-s + (−0.273 − 0.961i)10-s + (−0.850 + 0.526i)11-s + (0.850 − 0.526i)12-s + (−0.850 + 0.526i)13-s + (−0.0922 + 0.995i)14-s + (0.445 + 0.895i)15-s + (−0.850 − 0.526i)16-s + (0.739 − 0.673i)17-s + ⋯
L(s)  = 1  + (0.602 + 0.798i)2-s + (−0.739 − 0.673i)3-s + (−0.273 + 0.961i)4-s + (−0.932 − 0.361i)5-s + (0.0922 − 0.995i)6-s + (0.739 + 0.673i)7-s + (−0.932 + 0.361i)8-s + (0.0922 + 0.995i)9-s + (−0.273 − 0.961i)10-s + (−0.850 + 0.526i)11-s + (0.850 − 0.526i)12-s + (−0.850 + 0.526i)13-s + (−0.0922 + 0.995i)14-s + (0.445 + 0.895i)15-s + (−0.850 − 0.526i)16-s + (0.739 − 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0761 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0761 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4013\)
Sign: $-0.0761 + 0.997i$
Analytic conductor: \(18.6362\)
Root analytic conductor: \(18.6362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4013} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4013,\ (0:\ ),\ -0.0761 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8037092617 + 0.8674315607i\)
\(L(\frac12)\) \(\approx\) \(0.8037092617 + 0.8674315607i\)
\(L(1)\) \(\approx\) \(0.8364459023 + 0.3666129680i\)
\(L(1)\) \(\approx\) \(0.8364459023 + 0.3666129680i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4013 \( 1 \)
good2 \( 1 + (0.602 + 0.798i)T \)
3 \( 1 + (-0.739 - 0.673i)T \)
5 \( 1 + (-0.932 - 0.361i)T \)
7 \( 1 + (0.739 + 0.673i)T \)
11 \( 1 + (-0.850 + 0.526i)T \)
13 \( 1 + (-0.850 + 0.526i)T \)
17 \( 1 + (0.739 - 0.673i)T \)
19 \( 1 + (0.0922 + 0.995i)T \)
23 \( 1 + (-0.445 - 0.895i)T \)
29 \( 1 + (0.982 - 0.183i)T \)
31 \( 1 + (0.0922 - 0.995i)T \)
37 \( 1 + (-0.0922 - 0.995i)T \)
41 \( 1 + (0.445 + 0.895i)T \)
43 \( 1 + (0.739 - 0.673i)T \)
47 \( 1 + (0.932 - 0.361i)T \)
53 \( 1 + (-0.602 - 0.798i)T \)
59 \( 1 + (0.445 - 0.895i)T \)
61 \( 1 + (0.0922 + 0.995i)T \)
67 \( 1 + (-0.602 - 0.798i)T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (0.445 + 0.895i)T \)
79 \( 1 + (0.850 + 0.526i)T \)
83 \( 1 + (0.0922 - 0.995i)T \)
89 \( 1 + (-0.602 - 0.798i)T \)
97 \( 1 + (-0.602 + 0.798i)T \)
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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

−18.17792409327766058658993071128, −17.77921335107779023451598591614, −17.00374569163043868550047793907, −16.00707383711502690642249045457, −15.49432561962088513576053418121, −14.95210554536271066623250128644, −14.20779636567929892638852143400, −13.56526721164039748091001714818, −12.38653312592972296051020958976, −12.2007216840427470379344736529, −11.27337790402794655439034916400, −10.781034867234895365433151734737, −10.4270019312977103908175777134, −9.67599866464223086010635926452, −8.61185352646049188387265201538, −7.7741376381193012170085394291, −7.02111246878291800842439322848, −6.03860953645264420289698965080, −5.20889441368634400088423963614, −4.75186473889916830994014980911, −4.050481958057773375159346306642, −3.27485707881009565046395874651, −2.70204985069267762158584742378, −1.2529722134568944940246343587, −0.49378369422121991860883340256, 0.68183751278453550772563207782, 2.11393029567285265250865424343, 2.6805937933239371736884976881, 4.03123066877711130158924125389, 4.67539677427353850461066285685, 5.22592840367600988755399696083, 5.816614908618012115355668412967, 6.769822317386062334516086301041, 7.58061319623138425544716026657, 7.84066861741836116181478075729, 8.483408106772732696575562492294, 9.53307035830821302741243370961, 10.5651263272836343508787995716, 11.60094964124225930457416045162, 11.94104638798753524085469229087, 12.48035528567693486005819630385, 12.94090924430092338840990537661, 14.14657958111716760877882718206, 14.445748671026479272004513219187, 15.369573869251089798643328222059, 15.99105301720991765942094067953, 16.51058387268550926035959169524, 17.18306340188926835672965724162, 17.90841896924354764441432294888, 18.55843282371321892378151965673

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