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Properties

Label 2-1805-5.4-c1-0-60
Degree $2$
Conductor $1805$
Sign $-1$
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  + 2.79i·2-s + 1.12i·3-s − 5.83·4-s + 2.23·5-s − 3.14·6-s − 10.7i·8-s + 1.74·9-s + 6.25i·10-s + 2.92·11-s − 6.54i·12-s − 3.08i·13-s + 2.50i·15-s + 18.3·16-s + 4.87i·18-s − 13.0·20-s + ⋯
L(s)  = 1  + 1.97i·2-s + 0.647i·3-s − 2.91·4-s + 0.999·5-s − 1.28·6-s − 3.79i·8-s + 0.580·9-s + 1.97i·10-s + 0.883·11-s − 1.89i·12-s − 0.856i·13-s + 0.647i·15-s + 4.59·16-s + 1.14i·18-s − 2.91·20-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.899734977\)
\(L(\frac12)\) \(\approx\) \(1.899734977\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 2.23T \)
19 \( 1 \)
good2 \( 1 - 2.79iT - 2T^{2} \)
3 \( 1 - 1.12iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 2.92T + 11T^{2} \)
13 \( 1 + 3.08iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 9.15iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 13.6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 1.11T + 61T^{2} \)
67 \( 1 + 13.7iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−9.363652427233524720811363457384, −8.905998929631447424013479892588, −7.986722437009396608725954133215, −7.14287518287308231224348293729, −6.43447769117389613689827028465, −5.78598569640694145453500347630, −4.99171483108239091179592219480, −4.33846207334861297379204831632, −3.33146102065280175589905290480, −1.15261825081475647673729503567, 0.932849486335523003189705802122, 1.80680952843158242661251032438, 2.37190348813924003405344479745, 3.67087834470992182704411540939, 4.39501813760910027425630571552, 5.37181031918653889432451072013, 6.36701659474780854067603249111, 7.39038285785926112907607123699, 8.646697997335302124280432108242, 9.116485398965783502553432740543

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