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Properties

Label 2-1575-105.62-c0-0-3
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  + (0.366 − 0.366i)2-s + 0.732i·4-s + (−0.707 − 0.707i)7-s + (0.633 + 0.633i)8-s + 1.93i·11-s − 0.517·14-s − 0.267·16-s + (0.707 + 0.707i)22-s + (1.36 + 1.36i)23-s + (0.517 − 0.517i)28-s + 0.517·29-s + (−0.732 + 0.732i)32-s + (−0.707 − 0.707i)37-s + (0.707 − 0.707i)43-s − 1.41·44-s + ⋯
L(s)  = 1  + (0.366 − 0.366i)2-s + 0.732i·4-s + (−0.707 − 0.707i)7-s + (0.633 + 0.633i)8-s + 1.93i·11-s − 0.517·14-s − 0.267·16-s + (0.707 + 0.707i)22-s + (1.36 + 1.36i)23-s + (0.517 − 0.517i)28-s + 0.517·29-s + (−0.732 + 0.732i)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: $\chi_{1575} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ 0.679 - 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.236888347\)
\(L(\frac12)\) \(\approx\) \(1.236888347\)
\(L(1)\) 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 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
11 \( 1 - 1.93iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
29 \( 1 - 0.517T + T^{2} \)
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 \( 1 - 0.517iT - T^{2} \)
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

−9.733504672347671665072905386892, −9.100008071857990013277522640747, −7.970205960913710794190104157396, −7.09950950286914234861124215572, −6.95391048302716835516072821473, −5.37624061816045063627250995126, −4.52455558546362392579530879157, −3.79161089902089641625625424940, −2.88680937668118379592671039789, −1.74940398504564107309680385852, 0.921860638322947005572199690746, 2.63269229092988112320683725222, 3.48512638490808730413728705295, 4.76124582424633634832530299355, 5.50947716002468029369939036991, 6.33664940427641948292983789309, 6.63028795970419229790062518053, 7.980734504600801961077484527043, 8.848485231499468168517041575977, 9.333108473931088115774384910537

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