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Properties

Label 2-3800-5.4-c1-0-49
Degree $2$
Conductor $3800$
Sign $0.894 + 0.447i$
Analytic cond. $30.3431$
Root an. cond. $5.50846$
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  + 2i·3-s − 9-s − 4·11-s − 4i·13-s − 2i·17-s − 19-s + 4i·23-s + 4i·27-s + 6·29-s − 8·31-s − 8i·33-s − 4i·37-s + 8·39-s − 2·41-s − 4i·43-s + ⋯
L(s)  = 1  + 1.15i·3-s − 0.333·9-s − 1.20·11-s − 1.10i·13-s − 0.485i·17-s − 0.229·19-s + 0.834i·23-s + 0.769i·27-s + 1.11·29-s − 1.43·31-s − 1.39i·33-s − 0.657i·37-s + 1.28·39-s − 0.312·41-s − 0.609i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(30.3431\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (3649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.250727010\)
\(L(\frac12)\) \(\approx\) \(1.250727010\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 3T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 14iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 + 4iT - 97T^{2} \)
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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

−8.600490612133485140072493170118, −7.67473378577160993432617844906, −7.19083435255323682069450919347, −5.92802227178778727402680733423, −5.25087172880662677657622445678, −4.81891976734193592567706230130, −3.72339528297593534327338333426, −3.15642961271282818174487236214, −2.09514546165216426893649880899, −0.40693001706152089466805959728, 1.02663657397353437532918763041, 2.06889194092672978034692285573, 2.70418186520687253034719450080, 3.99008239393249547241255919867, 4.79337381446210053928085388874, 5.74021814745134973998248861979, 6.53129176987406667258668231547, 7.03026440298664264605205141397, 7.79429008931522625111058771492, 8.368594485502377344560929485343

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