Properties

Label 2-1575-5.4-c1-0-20
Degree $2$
Conductor $1575$
Sign $0.447 - 0.894i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
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  + 0.618i·2-s + 1.61·4-s + i·7-s + 2.23i·8-s + 0.236·11-s − 1.23i·13-s − 0.618·14-s + 1.85·16-s − 2.47i·17-s + 4.47·19-s + 0.145i·22-s + 6.23i·23-s + 0.763·26-s + 1.61i·28-s + 5·29-s + ⋯
L(s)  = 1  + 0.437i·2-s + 0.809·4-s + 0.377i·7-s + 0.790i·8-s + 0.0711·11-s − 0.342i·13-s − 0.165·14-s + 0.463·16-s − 0.599i·17-s + 1.02·19-s + 0.0311i·22-s + 1.30i·23-s + 0.149·26-s + 0.305i·28-s + 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.212430453\)
\(L(\frac12)\) \(\approx\) \(2.212430453\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - iT \)
good2 \( 1 - 0.618iT - 2T^{2} \)
11 \( 1 - 0.236T + 11T^{2} \)
13 \( 1 + 1.23iT - 13T^{2} \)
17 \( 1 + 2.47iT - 17T^{2} \)
19 \( 1 - 4.47T + 19T^{2} \)
23 \( 1 - 6.23iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 3.70T + 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 + 4.76T + 41T^{2} \)
43 \( 1 + 1.76iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 8.47iT - 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 9.70T + 61T^{2} \)
67 \( 1 + 4.23iT - 67T^{2} \)
71 \( 1 + 8.70T + 71T^{2} \)
73 \( 1 - 8.76iT - 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 7.70iT - 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 5.23iT - 97T^{2} \)
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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

−9.567004563453203908671643053701, −8.636687561997886927221941232280, −7.80238808186586014004096056136, −7.21488384261815347406170975871, −6.34914261091533120952771609799, −5.57302780989507468883094691115, −4.85346312977201946070230751135, −3.37440569280752008160819193899, −2.61856972290788079706781445846, −1.32307223486623837970566515950, 0.960025870051568793908766295894, 2.13901683207895419866799350972, 3.13507859693014896465873668768, 4.04248046867325165397879752589, 5.08670046521406669716881082846, 6.27806741095944333729665022090, 6.76091797294108284624859361916, 7.66458095957089264558556261364, 8.449300032768824011024544599625, 9.463265282722077811398941432539

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