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Properties

Label 2-1805-5.4-c1-0-105
Degree $2$
Conductor $1805$
Sign $0.335 + 0.941i$
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  + 0.244i·2-s − 2.73i·3-s + 1.94·4-s + (0.750 + 2.10i)5-s + 0.669·6-s − 1.94i·7-s + 0.963i·8-s − 4.49·9-s + (−0.514 + 0.183i)10-s + 4.23·11-s − 5.31i·12-s − 1.26i·13-s + 0.474·14-s + (5.76 − 2.05i)15-s + 3.64·16-s − 2.46i·17-s + ⋯
L(s)  = 1  + 0.172i·2-s − 1.58i·3-s + 0.970·4-s + (0.335 + 0.941i)5-s + 0.273·6-s − 0.733i·7-s + 0.340i·8-s − 1.49·9-s + (−0.162 + 0.0580i)10-s + 1.27·11-s − 1.53i·12-s − 0.352i·13-s + 0.126·14-s + (1.48 − 0.530i)15-s + 0.911·16-s − 0.596i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.335 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.335 + 0.941i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.335 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.549302274\)
\(L(\frac12)\) \(\approx\) \(2.549302274\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.750 - 2.10i)T \)
19 \( 1 \)
good2 \( 1 - 0.244iT - 2T^{2} \)
3 \( 1 + 2.73iT - 3T^{2} \)
7 \( 1 + 1.94iT - 7T^{2} \)
11 \( 1 - 4.23T + 11T^{2} \)
13 \( 1 + 1.26iT - 13T^{2} \)
17 \( 1 + 2.46iT - 17T^{2} \)
23 \( 1 - 5.13iT - 23T^{2} \)
29 \( 1 - 8.64T + 29T^{2} \)
31 \( 1 + 5.10T + 31T^{2} \)
37 \( 1 + 11.0iT - 37T^{2} \)
41 \( 1 + 2.49T + 41T^{2} \)
43 \( 1 + 4.46iT - 43T^{2} \)
47 \( 1 - 6.76iT - 47T^{2} \)
53 \( 1 - 0.689iT - 53T^{2} \)
59 \( 1 - 5.19T + 59T^{2} \)
61 \( 1 - 2.80T + 61T^{2} \)
67 \( 1 - 5.21iT - 67T^{2} \)
71 \( 1 + 0.791T + 71T^{2} \)
73 \( 1 + 1.41iT - 73T^{2} \)
79 \( 1 + 6.06T + 79T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + 2.23T + 89T^{2} \)
97 \( 1 - 8.60iT - 97T^{2} \)
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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

−8.973337129556075698297345736692, −7.87445484264563796208457624587, −7.17169669329360366644571226597, −7.01474483878418968625433374722, −6.22323701280248992002791491261, −5.57891552197894961037074455283, −3.80631682044334951562280363335, −2.84796949638608450558841382305, −1.94172428228637461971191002159, −1.04521685318679217359592371699, 1.40474105745471062963554632853, 2.58941971579092537476853390666, 3.67373156659570540063132648150, 4.44310392380080204564003483820, 5.24464445872476812458524127975, 6.13355545531824022951681658163, 6.74916237359673052632359692304, 8.381080524205293873975936832827, 8.679995086694173194403441939804, 9.583440212342008909795911772275

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