↑

Properties

Label 2-3724-931.303-c0-0-1
Degree $2$
Conductor $3724$
Sign $0.325 + 0.945i$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.658 − 1.67i)5-s + (−0.988 − 0.149i)7-s + (0.955 + 0.294i)9-s + (1.82 − 0.563i)11-s + (1.36 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (1.57 − 1.07i)23-s + (−1.64 + 1.52i)25-s + (0.400 + 1.75i)35-s + (−1.12 − 1.40i)43-s + (−0.134 − 1.79i)45-s + (−0.535 − 0.496i)47-s + (0.955 + 0.294i)49-s + (−2.14 − 2.69i)55-s + (−0.109 + 1.46i)61-s + ⋯
L(s)  = 1  + (−0.658 − 1.67i)5-s + (−0.988 − 0.149i)7-s + (0.955 + 0.294i)9-s + (1.82 − 0.563i)11-s + (1.36 + 0.930i)17-s + (−0.5 + 0.866i)19-s + (1.57 − 1.07i)23-s + (−1.64 + 1.52i)25-s + (0.400 + 1.75i)35-s + (−1.12 − 1.40i)43-s + (−0.134 − 1.79i)45-s + (−0.535 − 0.496i)47-s + (0.955 + 0.294i)49-s + (−2.14 − 2.69i)55-s + (−0.109 + 1.46i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (2165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.325 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.268498031\)
\(L(\frac12)\) \(\approx\) \(1.268498031\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.988 + 0.149i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.955 - 0.294i)T^{2} \)
5 \( 1 + (0.658 + 1.67i)T + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (-1.82 + 0.563i)T + (0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (-1.36 - 0.930i)T + (0.365 + 0.930i)T^{2} \)
23 \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (0.733 + 0.680i)T^{2} \)
61 \( 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.733 + 0.680i)T + (0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 - T^{2} \)
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

−8.741557554733924528691259994299, −7.949288400729124599760319530705, −7.06007636105039331587917744519, −6.36012734227503642895875553281, −5.52651980942087389578618612203, −4.60268169647647655558314582621, −3.88540942904998175243511463391, −3.47255376264008178031271292999, −1.58483494639346233134762855115, −0.925553352201926599715407459906, 1.28018722407742017471674057945, 2.75754281777982667396991546772, 3.40759393153548350317282677274, 3.92793913247470936542927677968, 4.98846370241939527548111207745, 6.32798233960809538214362156504, 6.75755488998173268999095769602, 7.09854815937167332526067952178, 7.80361326786672052848235755467, 9.122884254551185064407551800371

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