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Properties

Label 2-1900-95.49-c1-0-9
Degree $2$
Conductor $1900$
Sign $-0.393 - 0.919i$
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  + (0.306 + 0.176i)3-s + 4.30i·7-s + (−1.43 − 2.48i)9-s + 6.01·11-s + (−5.15 + 2.97i)13-s + (3.35 + 1.93i)17-s + (−4.19 − 1.17i)19-s + (−0.760 + 1.31i)21-s + (0.678 − 0.391i)23-s − 2.07i·27-s + (3.98 + 6.89i)29-s − 4.49·31-s + (1.84 + 1.06i)33-s + 0.988i·37-s − 2.10·39-s + ⋯
L(s)  = 1  + (0.176 + 0.102i)3-s + 1.62i·7-s + (−0.479 − 0.829i)9-s + 1.81·11-s + (−1.42 + 0.825i)13-s + (0.813 + 0.469i)17-s + (−0.963 − 0.268i)19-s + (−0.166 + 0.287i)21-s + (0.141 − 0.0816i)23-s − 0.399i·27-s + (0.739 + 1.28i)29-s − 0.806·31-s + (0.320 + 0.184i)33-s + 0.162i·37-s − 0.336·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.393 - 0.919i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.393 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.426596695\)
\(L(\frac12)\) \(\approx\) \(1.426596695\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.19 + 1.17i)T \)
good3 \( 1 + (-0.306 - 0.176i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 4.30iT - 7T^{2} \)
11 \( 1 - 6.01T + 11T^{2} \)
13 \( 1 + (5.15 - 2.97i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.35 - 1.93i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.678 + 0.391i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.98 - 6.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.49T + 31T^{2} \)
37 \( 1 - 0.988iT - 37T^{2} \)
41 \( 1 + (3.15 - 5.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.35 - 0.785i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.09 + 0.630i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.05 - 4.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.62 + 4.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.80 + 4.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.09 - 3.52i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.90 + 5.03i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.00 - 4.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.99 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.58iT - 83T^{2} \)
89 \( 1 + (1.69 + 2.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.40 + 3.69i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.207277625608143741133953790530, −8.964523347865291206854339859652, −8.139055667316825006669560836120, −6.78767910235482386730221211027, −6.43157695081682644513664622943, −5.49323123246256080812603484316, −4.56517638898403828805238356885, −3.54529267533513895286007414653, −2.61090476621726862029255961812, −1.55634334099081512195129026729, 0.50505165056096775413731380216, 1.79933681564721986642235693263, 3.07179541681942714895460267527, 4.03612684577890163530686892013, 4.71875104670412421829812249490, 5.78406136011097441031781228059, 6.80923763221398324607401503025, 7.43183649186068049424169164926, 7.994611988649325345530833533705, 8.981868388897100501460874465468

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