↑

Properties

Label 2-624-39.32-c1-0-0
Degree $2$
Conductor $624$
Sign $-0.911 + 0.410i$
Analytic cond. $4.98266$
Root an. cond. $2.23218$
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  + (0.0464 + 1.73i)3-s + (−1.33 − 1.33i)5-s + (−0.566 + 2.11i)7-s + (−2.99 + 0.160i)9-s + (−0.256 − 0.958i)11-s + (−3.27 + 1.51i)13-s + (2.25 − 2.38i)15-s + (−2.36 − 4.08i)17-s + (−1.44 − 0.386i)19-s + (−3.68 − 0.882i)21-s + (−1.71 + 2.96i)23-s − 1.41i·25-s + (−0.417 − 5.17i)27-s + (0.733 + 0.423i)29-s + (−3.66 + 3.66i)31-s + ⋯
L(s)  = 1  + (0.0268 + 0.999i)3-s + (−0.598 − 0.598i)5-s + (−0.214 + 0.798i)7-s + (−0.998 + 0.0535i)9-s + (−0.0774 − 0.288i)11-s + (−0.907 + 0.421i)13-s + (0.582 − 0.614i)15-s + (−0.572 − 0.991i)17-s + (−0.330 − 0.0886i)19-s + (−0.804 − 0.192i)21-s + (−0.357 + 0.618i)23-s − 0.283i·25-s + (−0.0803 − 0.996i)27-s + (0.136 + 0.0786i)29-s + (−0.657 + 0.657i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.911 + 0.410i$
Analytic conductor: \(4.98266\)
Root analytic conductor: \(2.23218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :1/2),\ -0.911 + 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0416268 - 0.193966i\)
\(L(\frac12)\) \(\approx\) \(0.0416268 - 0.193966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.0464 - 1.73i)T \)
13 \( 1 + (3.27 - 1.51i)T \)
good5 \( 1 + (1.33 + 1.33i)T + 5iT^{2} \)
7 \( 1 + (0.566 - 2.11i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.256 + 0.958i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.36 + 4.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.44 + 0.386i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.71 - 2.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.733 - 0.423i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.66 - 3.66i)T - 31iT^{2} \)
37 \( 1 + (10.5 - 2.82i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-8.93 + 2.39i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.05 - 3.49i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.384 - 0.384i)T - 47iT^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + (11.7 + 3.14i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.83 - 4.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.52 + 5.67i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.69 - 10.0i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.393 + 0.393i)T + 73iT^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + (2.25 + 2.25i)T + 83iT^{2} \)
89 \( 1 + (-4.60 - 17.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.28 - 2.21i)T + (84.0 + 48.5i)T^{2} \)
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

−11.14310640702528861115088999668, −10.13211508851761633142493510640, −9.216030090318452096481932115266, −8.791368880035875000269316111472, −7.78994790213931420026539163622, −6.55411637620744275221006811414, −5.30329863451029635457558911916, −4.73957696885031695873632950909, −3.61512702893655280465051698598, −2.43742745548251550272849595920, 0.10060860318097141847206337463, 1.97829843068474444134301922949, 3.23368582529065701193377666591, 4.34149221873511526244269055812, 5.77733457108236566562102238729, 6.79034293490645406862128695415, 7.38791064182515738378890639063, 8.057581278152330665912910323965, 9.130437622215161709454863027566, 10.43300256613566502874917670192

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