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Properties

Label 2-1575-105.83-c0-0-10
Degree $2$
Conductor $1575$
Sign $-0.583 + 0.812i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.583 + 0.812i$
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: \((2,\ 1575,\ (\ :0),\ -0.583 + 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8699717508\)
\(L(\frac12)\) \(\approx\) \(0.8699717508\)
\(L(1)\) 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

−9.429945262509394613711219070863, −8.517030376886790596985894533800, −8.042835155004543134045134854225, −6.93469527039154666004994314650, −5.85972655628513521156961662736, −5.46309950015762734860962019419, −4.21554889279392513614667198906, −3.24237934294312350315762593549, −1.91367649840646536981872745317, −0.790563722286526069708630232352, 1.92369650854773894545859383406, 2.79711425493672261724523300334, 4.27341831652717955657036709063, 4.71719847037648989923969670203, 6.04223698347803549483599558539, 6.82171107798779923581676283463, 7.67253603800835825057058636670, 8.193192952355617256424148355955, 8.978960158951648640334258790602, 9.751157767346415551629491751600

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