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Properties

Label 1-812-812.3-r1-0-0
Degree $1$
Conductor $812$
Sign $-0.989 + 0.146i$
Analytic cond. $87.2615$
Root an. cond. $87.2615$
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  + (−0.930 − 0.365i)3-s + (0.0747 − 0.997i)5-s + (0.733 + 0.680i)9-s + (−0.680 − 0.733i)11-s + (−0.222 − 0.974i)13-s + (−0.433 + 0.900i)15-s + (0.866 − 0.5i)17-s + (0.930 − 0.365i)19-s + (−0.826 + 0.563i)23-s + (−0.988 − 0.149i)25-s + (−0.433 − 0.900i)27-s + (0.563 − 0.826i)31-s + (0.365 + 0.930i)33-s + (0.680 − 0.733i)37-s + (−0.149 + 0.988i)39-s + ⋯
L(s)  = 1  + (−0.930 − 0.365i)3-s + (0.0747 − 0.997i)5-s + (0.733 + 0.680i)9-s + (−0.680 − 0.733i)11-s + (−0.222 − 0.974i)13-s + (−0.433 + 0.900i)15-s + (0.866 − 0.5i)17-s + (0.930 − 0.365i)19-s + (−0.826 + 0.563i)23-s + (−0.988 − 0.149i)25-s + (−0.433 − 0.900i)27-s + (0.563 − 0.826i)31-s + (0.365 + 0.930i)33-s + (0.680 − 0.733i)37-s + (−0.149 + 0.988i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $-0.989 + 0.146i$
Analytic conductor: \(87.2615\)
Root analytic conductor: \(87.2615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 812,\ (1:\ ),\ -0.989 + 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08292316568 - 1.123378013i\)
\(L(\frac12)\) \(\approx\) \(-0.08292316568 - 1.123378013i\)
\(L(1)\) \(\approx\) \(0.6765205960 - 0.4252831437i\)
\(L(1)\) \(\approx\) \(0.6765205960 - 0.4252831437i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.930 - 0.365i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
11 \( 1 + (-0.680 - 0.733i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.930 - 0.365i)T \)
23 \( 1 + (-0.826 + 0.563i)T \)
31 \( 1 + (0.563 - 0.826i)T \)
37 \( 1 + (0.680 - 0.733i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.433 - 0.900i)T \)
47 \( 1 + (-0.294 + 0.955i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.149 - 0.988i)T \)
67 \( 1 + (0.955 - 0.294i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.997 - 0.0747i)T \)
79 \( 1 + (0.680 - 0.733i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.997 + 0.0747i)T \)
97 \( 1 + (-0.781 - 0.623i)T \)
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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

−22.55246638786727885658471767857, −21.60978982728990000842395219126, −21.211794947130119090181382544618, −20.099035155740627480015445102684, −19.010100350770703626485110196306, −18.231641024478178773811929191591, −17.8438075446361710951870272070, −16.70771509244754263396711196564, −16.14083239780926132014764617918, −15.138460756264211685543756443159, −14.53313511746232592892307559864, −13.55040756976116501078577398736, −12.348723221069546892900462037789, −11.79670367583180708091433091975, −10.91110524722741955292999615757, −9.97158762391782152560494215923, −9.77150616300949565860811380536, −8.11338878551391375396109082762, −7.16968509899839421931269114085, −6.45006138629895174983192845667, −5.57605715170589922293498039653, −4.60337070258348845471531833483, −3.66662202745057978667205538250, −2.51031599972350954583460934827, −1.27594330761789465533091781795, 0.38043047348351788139841348067, 0.89379853858446677659396697887, 2.25046410711941556611746402327, 3.57989259145117069869207673425, 4.8918733246686262581187565560, 5.469299752157610773806386346573, 6.09073226693838353152703190658, 7.666554116600547102637983139, 7.85618555274279950334042310910, 9.26288743890260242270202880755, 10.082681315822589246588410420519, 11.00084988631390273455265763676, 11.918866224626027871050052378714, 12.50205285819134884148945440827, 13.36030696193714697827200469118, 13.96182077310453090493659916113, 15.58309909022801889859132610143, 15.99317013927629858609631026511, 16.830328108523202885119927340279, 17.55327110555618118136142182599, 18.26720064898220234885123765614, 19.100218265167705867232542660, 20.07815610529323823147865369365, 20.84180907404995391014859991652, 21.66754901743419900437118934830

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