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Properties

Label 2-1900-95.37-c1-0-22
Degree $2$
Conductor $1900$
Sign $0.301 + 0.953i$
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  + (0.814 + 0.814i)3-s + (−1.28 − 1.28i)7-s − 1.67i·9-s − 0.814·11-s + (2.02 + 2.02i)13-s + (−1.28 − 1.28i)17-s + (−4.09 − 1.48i)19-s − 2.10i·21-s + (1.75 − 1.75i)23-s + (3.80 − 3.80i)27-s + 1.12·29-s − 4.96i·31-s + (−0.663 − 0.663i)33-s + (3.80 − 3.80i)37-s + 3.30i·39-s + ⋯
L(s)  = 1  + (0.470 + 0.470i)3-s + (−0.487 − 0.487i)7-s − 0.557i·9-s − 0.245·11-s + (0.561 + 0.561i)13-s + (−0.312 − 0.312i)17-s + (−0.940 − 0.341i)19-s − 0.458i·21-s + (0.366 − 0.366i)23-s + (0.732 − 0.732i)27-s + 0.209·29-s − 0.891i·31-s + (−0.115 − 0.115i)33-s + (0.625 − 0.625i)37-s + 0.528i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.301 + 0.953i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.301 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.496206994\)
\(L(\frac12)\) \(\approx\) \(1.496206994\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.09 + 1.48i)T \)
good3 \( 1 + (-0.814 - 0.814i)T + 3iT^{2} \)
7 \( 1 + (1.28 + 1.28i)T + 7iT^{2} \)
11 \( 1 + 0.814T + 11T^{2} \)
13 \( 1 + (-2.02 - 2.02i)T + 13iT^{2} \)
17 \( 1 + (1.28 + 1.28i)T + 17iT^{2} \)
23 \( 1 + (-1.75 + 1.75i)T - 23iT^{2} \)
29 \( 1 - 1.12T + 29T^{2} \)
31 \( 1 + 4.96iT - 31T^{2} \)
37 \( 1 + (-3.80 + 3.80i)T - 37iT^{2} \)
41 \( 1 + 5.22iT - 41T^{2} \)
43 \( 1 + (1.22 - 1.22i)T - 43iT^{2} \)
47 \( 1 + (4.40 + 4.40i)T + 47iT^{2} \)
53 \( 1 + (-5.41 - 5.41i)T + 53iT^{2} \)
59 \( 1 + 3.99T + 59T^{2} \)
61 \( 1 + 6.46T + 61T^{2} \)
67 \( 1 + (-6.22 + 6.22i)T - 67iT^{2} \)
71 \( 1 + 13.0iT - 71T^{2} \)
73 \( 1 + (0.985 - 0.985i)T - 73iT^{2} \)
79 \( 1 + 6.19T + 79T^{2} \)
83 \( 1 + (-5.56 + 5.56i)T - 83iT^{2} \)
89 \( 1 - 2.10T + 89T^{2} \)
97 \( 1 + (-3.65 + 3.65i)T - 97iT^{2} \)
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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

−9.052289749450781360877570013847, −8.521416339047826322715372990216, −7.46897803810829911294576780672, −6.62960262384587683835161972245, −6.04874682543706530620526592582, −4.72266314211444042971397552663, −4.03224310121559076021354324276, −3.25053266603270335531693858899, −2.17676507093246553307476149394, −0.51589897591401449708658617807, 1.41126425797466328307189233552, 2.52395328169827531011522898332, 3.26410191465212376024518856486, 4.46026775451362695603470111720, 5.43370721635499240276027691968, 6.26892869500318690657501820233, 7.00839961921687725470380809366, 8.051670026244753807439657883723, 8.391063757282857390482346626679, 9.222959950899883132819905170764

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