↑

Properties

Label 2-380-95.37-c1-0-3
Degree $2$
Conductor $380$
Sign $0.925 + 0.378i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.63 − 1.63i)3-s + (2.17 + 0.539i)5-s + (2.17 + 2.17i)7-s + 2.36i·9-s − 1.70·11-s + (4.43 + 4.43i)13-s + (−2.67 − 4.43i)15-s + (−1.63 − 1.63i)17-s + (2.80 − 3.34i)19-s − 7.11i·21-s + (3.24 − 3.24i)23-s + (4.41 + 2.34i)25-s + (−1.03 + 1.03i)27-s + 3.27·29-s − 7.11i·31-s + ⋯
L(s)  = 1  + (−0.945 − 0.945i)3-s + (0.970 + 0.241i)5-s + (0.820 + 0.820i)7-s + 0.789i·9-s − 0.515·11-s + (1.23 + 1.23i)13-s + (−0.689 − 1.14i)15-s + (−0.395 − 0.395i)17-s + (0.642 − 0.766i)19-s − 1.55i·21-s + (0.677 − 0.677i)23-s + (0.883 + 0.468i)25-s + (−0.198 + 0.198i)27-s + 0.608·29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.925 + 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27533 - 0.250374i\)
\(L(\frac12)\) \(\approx\) \(1.27533 - 0.250374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 0.539i)T \)
19 \( 1 + (-2.80 + 3.34i)T \)
good3 \( 1 + (1.63 + 1.63i)T + 3iT^{2} \)
7 \( 1 + (-2.17 - 2.17i)T + 7iT^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 + (-4.43 - 4.43i)T + 13iT^{2} \)
17 \( 1 + (1.63 + 1.63i)T + 17iT^{2} \)
23 \( 1 + (-3.24 + 3.24i)T - 23iT^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 + 7.11iT - 31T^{2} \)
37 \( 1 + (0.128 - 0.128i)T - 37iT^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + (5.24 - 5.24i)T - 43iT^{2} \)
47 \( 1 + (-0.908 - 0.908i)T + 47iT^{2} \)
53 \( 1 + (-5.47 - 5.47i)T + 53iT^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 2.63T + 61T^{2} \)
67 \( 1 + (7.71 - 7.71i)T - 67iT^{2} \)
71 \( 1 - 1.51iT - 71T^{2} \)
73 \( 1 + (8.70 - 8.70i)T - 73iT^{2} \)
79 \( 1 - 4.78T + 79T^{2} \)
83 \( 1 + (-6.46 + 6.46i)T - 83iT^{2} \)
89 \( 1 + 5.34T + 89T^{2} \)
97 \( 1 + (-10.2 + 10.2i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.39511765792084812321927244157, −10.74925711476142792430106066193, −9.345483447385445660567236283322, −8.604121336807545715937991080887, −7.25290172002498588240957195217, −6.40999682217015452662405365810, −5.70590203172855734923246645774, −4.73372644464288662985161250618, −2.53841615455445987409703498804, −1.38248255726042983125528942120, 1.30256348673225848390441925455, 3.44466088670721926234269166298, 4.80025334811021321761709981501, 5.40474687237378891927351795710, 6.29357314760571044108970272940, 7.75552342440168818137872530743, 8.740887265626114247802569615475, 9.982895036403603320994719981214, 10.60853974622839189182927527852, 10.95067495509264631345993407192

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