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Properties

Label 2-1900-5.4-c1-0-13
Degree $2$
Conductor $1900$
Sign $0.894 - 0.447i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
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  + 2.91i·3-s − 2.91i·7-s − 5.48·9-s + 0.598·11-s − 2.56i·13-s − 2.91i·17-s + 19-s + 8.48·21-s − 1.16i·23-s − 7.22i·27-s + 1.40·29-s + 7.88·31-s + 1.74i·33-s + 6.53i·37-s + 7.48·39-s + ⋯
L(s)  = 1  + 1.68i·3-s − 1.10i·7-s − 1.82·9-s + 0.180·11-s − 0.712i·13-s − 0.706i·17-s + 0.229·19-s + 1.85·21-s − 0.243i·23-s − 1.39i·27-s + 0.260·29-s + 1.41·31-s + 0.303i·33-s + 1.07i·37-s + 1.19·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.633823095\)
\(L(\frac12)\) \(\approx\) \(1.633823095\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 2.91iT - 3T^{2} \)
7 \( 1 + 2.91iT - 7T^{2} \)
11 \( 1 - 0.598T + 11T^{2} \)
13 \( 1 + 2.56iT - 13T^{2} \)
17 \( 1 + 2.91iT - 17T^{2} \)
23 \( 1 + 1.16iT - 23T^{2} \)
29 \( 1 - 1.40T + 29T^{2} \)
31 \( 1 - 7.88T + 31T^{2} \)
37 \( 1 - 6.53iT - 37T^{2} \)
41 \( 1 - 9.48T + 41T^{2} \)
43 \( 1 + 0.824iT - 43T^{2} \)
47 \( 1 + 6.05iT - 47T^{2} \)
53 \( 1 + 0.824iT - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 1.88T + 61T^{2} \)
67 \( 1 + 5.11iT - 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 - 1.03iT - 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 15.7iT - 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + 7.70iT - 97T^{2} \)
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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

−9.467522992760864681067558146237, −8.654512292180004960640068901331, −7.84792874579816080421552443251, −6.90697217506277385600943806935, −5.87845858810740159772375331239, −4.93760497689470720930101600015, −4.38338206264045259287204692287, −3.56567199478877980048080237788, −2.75672072884568304880294799240, −0.73166792921442394998034599820, 1.06571583594025665855920872741, 2.11088558962344711655841084428, 2.77880378197640330873713521909, 4.18645507695309889330181886387, 5.50075200772429724884822699809, 6.10781246426943486383822099228, 6.75590242278276648983720880824, 7.57259679210902734514313811213, 8.303548196972688416876386102904, 8.902218427466237222353441082744

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