↑

Properties

Label 2-2023-119.104-c0-0-10
Degree $2$
Conductor $2023$
Sign $-0.950 + 0.311i$
Analytic cond. $1.00960$
Root an. cond. $1.00479$
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  + (−0.437 − 0.437i)2-s + (−0.236 − 0.570i)3-s − 0.618i·4-s + (1.49 − 0.619i)5-s + (−0.146 + 0.352i)6-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s + (0.437 − 0.437i)9-s + (−0.923 − 0.382i)10-s + (−0.352 + 0.146i)12-s + (0.236 + 0.570i)14-s + (−0.707 − 0.707i)15-s − 0.381·18-s + (−0.382 − 0.923i)20-s + 0.618i·21-s + ⋯
L(s)  = 1  + (−0.437 − 0.437i)2-s + (−0.236 − 0.570i)3-s − 0.618i·4-s + (1.49 − 0.619i)5-s + (−0.146 + 0.352i)6-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s + (0.437 − 0.437i)9-s + (−0.923 − 0.382i)10-s + (−0.352 + 0.146i)12-s + (0.236 + 0.570i)14-s + (−0.707 − 0.707i)15-s − 0.381·18-s + (−0.382 − 0.923i)20-s + 0.618i·21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $-0.950 + 0.311i$
Analytic conductor: \(1.00960\)
Root analytic conductor: \(1.00479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2023} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2023,\ (\ :0),\ -0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9368568069\)
\(L(\frac12)\) \(\approx\) \(0.9368568069\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.923 + 0.382i)T \)
17 \( 1 \)
good2 \( 1 + (0.437 + 0.437i)T + iT^{2} \)
3 \( 1 + (0.236 + 0.570i)T + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-1.49 + 0.619i)T + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.619 + 1.49i)T + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.570 + 0.236i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.437 - 0.437i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.14 - 1.14i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-0.570 - 0.236i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 - 1.61T + T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.570 - 0.236i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1.49 - 0.619i)T + (0.707 - 0.707i)T^{2} \)
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

−9.365964947210945540200797353026, −8.512044090782674629254088016791, −7.24493621044750348236391985809, −6.44275776537239965027472491940, −5.92684060737718548042745114695, −5.26532890421958706937709699462, −4.01916140553604377471644409184, −2.59702849389113293524884137330, −1.72145998117484053334014185998, −0.800771422075051687011462749243, 1.98305549532134175995055491109, 2.97574086428205995235584109483, 3.79251688470803308202502084718, 5.13345096390783903030912104986, 5.79811486257139269851495435738, 6.79548138695926048248609511386, 6.98003284499355560486203339997, 8.271738724102125015753786728225, 9.056487888666011982088768006398, 9.707115295227921203668870762597

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