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Properties

Label 2-1900-95.37-c1-0-28
Degree $2$
Conductor $1900$
Sign $-0.103 - 0.994i$
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  + (−1.79 − 1.79i)3-s + (−3.30 − 3.30i)7-s + 3.44i·9-s + 2.44·11-s + (−2.60 − 2.60i)13-s + (−1.48 − 1.48i)17-s + (−2.66 − 3.44i)19-s + 11.8i·21-s + (4.78 − 4.78i)23-s + (0.807 − 0.807i)27-s − 5.32·29-s − 6.52i·31-s + (−4.39 − 4.39i)33-s + (−7.00 + 7.00i)37-s + 9.34i·39-s + ⋯
L(s)  = 1  + (−1.03 − 1.03i)3-s + (−1.24 − 1.24i)7-s + 1.14i·9-s + 0.738·11-s + (−0.721 − 0.721i)13-s + (−0.359 − 0.359i)17-s + (−0.611 − 0.791i)19-s + 2.58i·21-s + (0.997 − 0.997i)23-s + (0.155 − 0.155i)27-s − 0.989·29-s − 1.17i·31-s + (−0.765 − 0.765i)33-s + (−1.15 + 1.15i)37-s + 1.49i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.103 - 0.994i$
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: \((2,\ 1900,\ (\ :1/2),\ -0.103 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3148266695\)
\(L(\frac12)\) \(\approx\) \(0.3148266695\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.66 + 3.44i)T \)
good3 \( 1 + (1.79 + 1.79i)T + 3iT^{2} \)
7 \( 1 + (3.30 + 3.30i)T + 7iT^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + (2.60 + 2.60i)T + 13iT^{2} \)
17 \( 1 + (1.48 + 1.48i)T + 17iT^{2} \)
23 \( 1 + (-4.78 + 4.78i)T - 23iT^{2} \)
29 \( 1 + 5.32T + 29T^{2} \)
31 \( 1 + 6.52iT - 31T^{2} \)
37 \( 1 + (7.00 - 7.00i)T - 37iT^{2} \)
41 \( 1 + 11.8iT - 41T^{2} \)
43 \( 1 + (1.81 - 1.81i)T - 43iT^{2} \)
47 \( 1 + (-3.30 - 3.30i)T + 47iT^{2} \)
53 \( 1 + (-3.41 - 3.41i)T + 53iT^{2} \)
59 \( 1 - 6.52T + 59T^{2} \)
61 \( 1 - 5.34T + 61T^{2} \)
67 \( 1 + (-4.58 + 4.58i)T - 67iT^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (2.96 - 2.96i)T - 73iT^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (-6.93 + 6.93i)T - 83iT^{2} \)
89 \( 1 + 5.32T + 89T^{2} \)
97 \( 1 + (3.77 - 3.77i)T - 97iT^{2} \)
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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

−8.624017298844084578458257288134, −7.26608217796233955471850834922, −7.10423804326120403493510871676, −6.46626531151551046309513811854, −5.63396355609552784880564098142, −4.59893101405742849098365538810, −3.61041300307506348912615793576, −2.41511160778105616630060345241, −0.882604794986435938474625663670, −0.17145753938675597170757850983, 1.95911235214491340601538738872, 3.31241019623486056810438414839, 4.04732095433375738086233352924, 5.11996807203680917589239368146, 5.68854382029181642084633005742, 6.45336250502893091988108805777, 7.09036563307271985448088460747, 8.625931861404578655895967831630, 9.207391001721710006825918381950, 9.750401554494658777316500024809

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