Properties

Label 2-1805-5.4-c1-0-34
Degree $2$
Conductor $1805$
Sign $-0.872 + 0.489i$
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  + 1.24i·2-s + 1.51i·3-s + 0.439·4-s + (−1.95 + 1.09i)5-s − 1.89·6-s + 0.568i·7-s + 3.04i·8-s + 0.710·9-s + (−1.36 − 2.43i)10-s + 3.18·11-s + 0.664i·12-s + 2.28i·13-s − 0.710·14-s + (−1.65 − 2.95i)15-s − 2.92·16-s + 2.21i·17-s + ⋯
L(s)  = 1  + 0.883i·2-s + 0.873i·3-s + 0.219·4-s + (−0.872 + 0.489i)5-s − 0.771·6-s + 0.214i·7-s + 1.07i·8-s + 0.236·9-s + (−0.432 − 0.770i)10-s + 0.961·11-s + 0.191i·12-s + 0.632i·13-s − 0.189·14-s + (−0.427 − 0.761i)15-s − 0.732·16-s + 0.537i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.872 + 0.489i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.872 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.666634932\)
\(L(\frac12)\) \(\approx\) \(1.666634932\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.95 - 1.09i)T \)
19 \( 1 \)
good2 \( 1 - 1.24iT - 2T^{2} \)
3 \( 1 - 1.51iT - 3T^{2} \)
7 \( 1 - 0.568iT - 7T^{2} \)
11 \( 1 - 3.18T + 11T^{2} \)
13 \( 1 - 2.28iT - 13T^{2} \)
17 \( 1 - 2.21iT - 17T^{2} \)
23 \( 1 + 0.0101iT - 23T^{2} \)
29 \( 1 + 1.14T + 29T^{2} \)
31 \( 1 + 9.12T + 31T^{2} \)
37 \( 1 - 8.97iT - 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 7.91iT - 43T^{2} \)
47 \( 1 + 12.5iT - 47T^{2} \)
53 \( 1 + 7.43iT - 53T^{2} \)
59 \( 1 + 8.46T + 59T^{2} \)
61 \( 1 + 7.36T + 61T^{2} \)
67 \( 1 - 3.51iT - 67T^{2} \)
71 \( 1 - 4.56T + 71T^{2} \)
73 \( 1 + 9.66iT - 73T^{2} \)
79 \( 1 - 7.97T + 79T^{2} \)
83 \( 1 + 13.7iT - 83T^{2} \)
89 \( 1 - 3.41T + 89T^{2} \)
97 \( 1 - 3.01iT - 97T^{2} \)
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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

−9.534595419932894873219296724924, −8.890361177013045937444743899025, −8.043296589653850339654844053225, −7.23677364308880639579837833501, −6.67146858466634651838803536618, −5.86509262964001031855688078693, −4.77983555060237477635605773157, −4.05372320401762924652178827439, −3.23138131997406599593125011156, −1.81056238203042925938265205686, 0.64535119696846202222365617505, 1.44708930137035250608533234822, 2.56888021524620981223745802899, 3.73386832878464874188591959560, 4.25690251714681679046111570870, 5.63444607267453333806751247232, 6.62444202606722251823512364589, 7.50577248718131287965236096793, 7.62775154805388092989579927230, 9.059985559317310797052756350556

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