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Properties

Label 2-1575-105.83-c0-0-7
Degree $2$
Conductor $1575$
Sign $-0.679 - 0.733i$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
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  + (1.36 + 1.36i)2-s + 2.73i·4-s + (0.707 − 0.707i)7-s + (−2.36 + 2.36i)8-s + 0.517i·11-s + 1.93·14-s − 3.73·16-s + (−0.707 + 0.707i)22-s + (0.366 − 0.366i)23-s + (1.93 + 1.93i)28-s − 1.93·29-s + (−2.73 − 2.73i)32-s + (0.707 − 0.707i)37-s + (−0.707 − 0.707i)43-s − 1.41·44-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)2-s + 2.73i·4-s + (0.707 − 0.707i)7-s + (−2.36 + 2.36i)8-s + 0.517i·11-s + 1.93·14-s − 3.73·16-s + (−0.707 + 0.707i)22-s + (0.366 − 0.366i)23-s + (1.93 + 1.93i)28-s − 1.93·29-s + (−2.73 − 2.73i)32-s + (0.707 − 0.707i)37-s + (−0.707 − 0.707i)43-s − 1.41·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.679 - 0.733i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.679 - 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.268912229\)
\(L(\frac12)\) \(\approx\) \(2.268912229\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
11 \( 1 - 0.517iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
29 \( 1 + 1.93T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
71 \( 1 - 1.93iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.740726624498240996135362841153, −8.684305056269299596251592458338, −7.971603878632795302280240432614, −7.24524159390452836084973253421, −6.79566737916180409757166308985, −5.67263000643270979491281114968, −5.08023500784039180141250674472, −4.20109767892290615997803522881, −3.60628599443616333668334704571, −2.22215545205281621639615463225, 1.38043885555316304327893211942, 2.35368353886260169942350334246, 3.26222762422419485066861486374, 4.16110148198179180010443735116, 5.08567586774935185413730278234, 5.64085880659797477565570340569, 6.43259580932900426961468598261, 7.72383849448969005786842312199, 8.908736332295660907067682774952, 9.513853471459289408040534317297

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