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Properties

Label 2-1575-105.62-c0-0-11
Degree $2$
Conductor $1575$
Sign $-0.960 + 0.279i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.960 + 0.279i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ -0.960 + 0.279i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.980229585\)
\(L(\frac12)\) \(\approx\) \(1.980229585\)
\(L(1)\) 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

−9.659617542562259772666237424190, −8.871984234763550180900498289606, −7.32827075700686506235908073756, −6.55951290705072074061209802903, −5.72564430960741932524276757402, −4.77575710658024135290157528146, −4.08300395660795834046983839454, −3.25229218228598211425576352289, −2.37974866495687844995343642332, −1.07449077360856215291491042836, 2.66662632293459873222333966248, 3.30058867738510469343717204354, 4.36954200036222743752555323892, 5.17726298766320299999509288139, 5.97581648799126839633159671506, 6.54368144537349932900432740506, 7.23694736932721308572798897650, 8.372085176315337754841016083677, 8.646377078709697428821468651407, 9.760351999705487895009310349328

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