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Properties

Label 2-1805-5.4-c1-0-19
Degree $2$
Conductor $1805$
Sign $0.914 + 0.403i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.96i·2-s + 3.10i·3-s − 1.87·4-s + (−2.04 − 0.902i)5-s − 6.10·6-s + 2.84i·7-s + 0.255i·8-s − 6.62·9-s + (1.77 − 4.02i)10-s − 0.295·11-s − 5.80i·12-s + 2.61i·13-s − 5.59·14-s + (2.80 − 6.34i)15-s − 4.24·16-s + 7.09i·17-s + ⋯
L(s)  = 1  + 1.39i·2-s + 1.79i·3-s − 0.935·4-s + (−0.914 − 0.403i)5-s − 2.49·6-s + 1.07i·7-s + 0.0904i·8-s − 2.20·9-s + (0.561 − 1.27i)10-s − 0.0891·11-s − 1.67i·12-s + 0.726i·13-s − 1.49·14-s + (0.723 − 1.63i)15-s − 1.06·16-s + 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.914 + 0.403i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 0.914 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9661289369\)
\(L(\frac12)\) \(\approx\) \(0.9661289369\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.04 + 0.902i)T \)
19 \( 1 \)
good2 \( 1 - 1.96iT - 2T^{2} \)
3 \( 1 - 3.10iT - 3T^{2} \)
7 \( 1 - 2.84iT - 7T^{2} \)
11 \( 1 + 0.295T + 11T^{2} \)
13 \( 1 - 2.61iT - 13T^{2} \)
17 \( 1 - 7.09iT - 17T^{2} \)
23 \( 1 + 2.66iT - 23T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + 1.74T + 31T^{2} \)
37 \( 1 + 0.722iT - 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 4.02iT - 43T^{2} \)
47 \( 1 - 2.94iT - 47T^{2} \)
53 \( 1 - 6.98iT - 53T^{2} \)
59 \( 1 - 8.84T + 59T^{2} \)
61 \( 1 - 6.62T + 61T^{2} \)
67 \( 1 - 1.93iT - 67T^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 - 3.05iT - 73T^{2} \)
79 \( 1 + 8.06T + 79T^{2} \)
83 \( 1 - 13.8iT - 83T^{2} \)
89 \( 1 + 0.551T + 89T^{2} \)
97 \( 1 + 6.60iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.647401768018907330618434178279, −8.896261050641072507621682543463, −8.557553247990617935050889432676, −7.905897898779127673809379712383, −6.69076205761881642647903345486, −5.77048466681879299093384555108, −5.28947222549819724501358760361, −4.32627264280267077715005768145, −3.90726235914782008766432699283, −2.51547647605495985868180261972, 0.43992828745667645433170694850, 1.00113713789487724328389053992, 2.35391147889967309754566936944, 3.04913661453041155435226341696, 3.89795409380461363214272818903, 5.12309365536705136151505086283, 6.48139726091572628636053961492, 7.21170587320951458058187994797, 7.54014427809725378113654945715, 8.382144135187216178105541033290

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