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Properties

Label 2-2023-119.104-c0-0-3
Degree $2$
Conductor $2023$
Sign $0.0883 + 0.996i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.382 − 0.923i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (1.84 + 0.765i)11-s + (−0.923 + 0.382i)14-s + 1.00·16-s − 1.00·18-s + (−0.765 − 1.84i)22-s + (0.923 + 0.382i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)29-s + (0.923 − 0.382i)37-s + (−0.707 + 0.707i)43-s + (−0.382 − 0.923i)46-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.382 − 0.923i)7-s + (−0.707 + 0.707i)8-s + (0.707 − 0.707i)9-s + (1.84 + 0.765i)11-s + (−0.923 + 0.382i)14-s + 1.00·16-s − 1.00·18-s + (−0.765 − 1.84i)22-s + (0.923 + 0.382i)23-s + (−0.707 + 0.707i)25-s + (0.382 + 0.923i)29-s + (0.923 − 0.382i)37-s + (−0.707 + 0.707i)43-s + (−0.382 − 0.923i)46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0883 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0883 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2023\)    =    \(7 \cdot 17^{2}\)
Sign: $0.0883 + 0.996i$
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: \((2,\ 2023,\ (\ :0),\ 0.0883 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9661369087\)
\(L(\frac12)\) \(\approx\) \(0.9661369087\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 \)
good2 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-1.84 - 0.765i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.923 - 0.382i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.382 - 0.923i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.923 + 0.382i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + (-0.923 + 0.382i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.923 + 0.382i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.440888415987589787477036946425, −8.710623144643989709636644998879, −7.58573656518475483836389561599, −6.88967376876402572895513532821, −6.21113355212129663578273647691, −4.92732032366149855920117964094, −4.08940856918823806726180056215, −3.26989821514281437530566510572, −1.63414425192825617163656292620, −1.20016023370048981756028957924, 1.31052656607063219621962303672, 2.66018212906417337310665502048, 3.83323549652810475577779806078, 4.67759018749887159997732209863, 5.93225777462942402000030949329, 6.42332505830169799289606616637, 7.25677124115379301922496081660, 8.120243039050006845658217789735, 8.623055764698917490286944725755, 9.295210370885252608384589146987

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