Properties

Label 2-45e2-5.4-c1-0-30
Degree $2$
Conductor $2025$
Sign $0.447 - 0.894i$
Analytic cond. $16.1697$
Root an. cond. $4.02115$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 4-s − 3i·7-s + 3i·8-s − 2·11-s + 2i·13-s + 3·14-s − 16-s + 4i·17-s + 8·19-s − 2i·22-s − 3i·23-s − 2·26-s − 3i·28-s + 29-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.5·4-s − 1.13i·7-s + 1.06i·8-s − 0.603·11-s + 0.554i·13-s + 0.801·14-s − 0.250·16-s + 0.970i·17-s + 1.83·19-s − 0.426i·22-s − 0.625i·23-s − 0.392·26-s − 0.566i·28-s + 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(16.1697\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.141060383\)
\(L(\frac12)\) \(\approx\) \(2.141060383\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - iT - 2T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + 3iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 14T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.216671908468601068603865209808, −8.142337194185448615514503784285, −7.64369990263612032540219959141, −7.01427774480794209098530282911, −6.25428291419858248444103050468, −5.43454586698286757171213198556, −4.52654511275193127997975982567, −3.51554734338535868413404096591, −2.42156515848049926800147770809, −1.12932340481397501055521220643, 0.898631429461865569707751739729, 2.26407020116329291878464938219, 2.87682520648639803868689590344, 3.70198749896840164476878621518, 5.30118154035997702690623498826, 5.48384601331890464584641640781, 6.74887922406124275868822580062, 7.43630468478314830095915477430, 8.246952852955722579449137054363, 9.262100517473119846947611803145

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