Properties

Label 2-380-380.7-c1-0-11
Degree $2$
Conductor $380$
Sign $-0.986 - 0.164i$
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  + (−0.824 + 1.14i)2-s + (−0.320 + 1.19i)3-s + (−0.640 − 1.89i)4-s + (1.11 + 1.93i)5-s + (−1.11 − 1.35i)6-s + (−1.23 + 1.23i)7-s + (2.70 + 0.826i)8-s + (1.26 + 0.732i)9-s + (−3.14 − 0.310i)10-s − 0.393i·11-s + (2.47 − 0.158i)12-s + (0.619 + 2.31i)13-s + (−0.401 − 2.44i)14-s + (−2.67 + 0.718i)15-s + (−3.17 + 2.42i)16-s + (−0.318 + 1.19i)17-s + ⋯
L(s)  = 1  + (−0.583 + 0.812i)2-s + (−0.185 + 0.690i)3-s + (−0.320 − 0.947i)4-s + (0.500 + 0.865i)5-s + (−0.453 − 0.553i)6-s + (−0.467 + 0.467i)7-s + (0.956 + 0.292i)8-s + (0.422 + 0.244i)9-s + (−0.995 − 0.0980i)10-s − 0.118i·11-s + (0.713 − 0.0458i)12-s + (0.171 + 0.640i)13-s + (−0.107 − 0.652i)14-s + (−0.690 + 0.185i)15-s + (−0.794 + 0.606i)16-s + (−0.0773 + 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.986 - 0.164i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.986 - 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0712813 + 0.862979i\)
\(L(\frac12)\) \(\approx\) \(0.0712813 + 0.862979i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.824 - 1.14i)T \)
5 \( 1 + (-1.11 - 1.93i)T \)
19 \( 1 + (4.15 + 1.32i)T \)
good3 \( 1 + (0.320 - 1.19i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.23 - 1.23i)T - 7iT^{2} \)
11 \( 1 + 0.393iT - 11T^{2} \)
13 \( 1 + (-0.619 - 2.31i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.318 - 1.19i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-1.65 + 0.443i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.84 + 1.06i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.76iT - 31T^{2} \)
37 \( 1 + (-1.95 - 1.95i)T + 37iT^{2} \)
41 \( 1 + (2.87 + 4.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.727 - 2.71i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.56 - 5.84i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.72 + 13.8i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.64 - 2.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.02 + 1.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.61 - 9.74i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-11.2 + 6.49i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.35 - 2.23i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.48 - 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.90 - 6.90i)T + 83iT^{2} \)
89 \( 1 + (-7.23 - 4.17i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.10 + 7.83i)T + (-84.0 - 48.5i)T^{2} \)
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

−11.25114836775868329096449464563, −10.70280387163459679083498661206, −9.751315207179755312563503022955, −9.287155255181882227730115243984, −8.092592668825762530741369110850, −6.88563412193042645566772687854, −6.25436845730813768183024724564, −5.20634113047785353035114390897, −3.97290117446328380918417858314, −2.14486747574604320848709552669, 0.73320991399964159650194245583, 1.96710637428772118313959726043, 3.60048729408761825184199491324, 4.82666424015444108498015025658, 6.29585622019697482763894976544, 7.30675619666304311947250069808, 8.282161313158859199596664889842, 9.213685419253475839395414433256, 10.02073688206387515628544029891, 10.79465147623442304181102450424

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