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Properties

Label 2-1575-15.2-c1-0-15
Degree $2$
Conductor $1575$
Sign $0.662 - 0.749i$
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  + (−1.96 + 1.96i)2-s − 5.71i·4-s + (−0.707 − 0.707i)7-s + (7.29 + 7.29i)8-s + 0.248i·11-s + (3.37 − 3.37i)13-s + 2.77·14-s − 17.2·16-s + (−1.75 + 1.75i)17-s + 3.91i·19-s + (−0.487 − 0.487i)22-s + (2.22 + 2.22i)23-s + 13.2i·26-s + (−4.04 + 4.04i)28-s + 2.65·29-s + ⋯
L(s)  = 1  + (−1.38 + 1.38i)2-s − 2.85i·4-s + (−0.267 − 0.267i)7-s + (2.58 + 2.58i)8-s + 0.0749i·11-s + (0.935 − 0.935i)13-s + 0.742·14-s − 4.31·16-s + (−0.426 + 0.426i)17-s + 0.898i·19-s + (−0.104 − 0.104i)22-s + (0.463 + 0.463i)23-s + 2.59i·26-s + (−0.763 + 0.763i)28-s + 0.492·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7404464708\)
\(L(\frac12)\) \(\approx\) \(0.7404464708\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (1.96 - 1.96i)T - 2iT^{2} \)
11 \( 1 - 0.248iT - 11T^{2} \)
13 \( 1 + (-3.37 + 3.37i)T - 13iT^{2} \)
17 \( 1 + (1.75 - 1.75i)T - 17iT^{2} \)
19 \( 1 - 3.91iT - 19T^{2} \)
23 \( 1 + (-2.22 - 2.22i)T + 23iT^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
37 \( 1 + (1.76 + 1.76i)T + 37iT^{2} \)
41 \( 1 + 7.42iT - 41T^{2} \)
43 \( 1 + (-0.716 + 0.716i)T - 43iT^{2} \)
47 \( 1 + (3.34 - 3.34i)T - 47iT^{2} \)
53 \( 1 + (4.25 + 4.25i)T + 53iT^{2} \)
59 \( 1 - 4.88T + 59T^{2} \)
61 \( 1 - 9.55T + 61T^{2} \)
67 \( 1 + (-5.98 - 5.98i)T + 67iT^{2} \)
71 \( 1 + 6.31iT - 71T^{2} \)
73 \( 1 + (-10.3 + 10.3i)T - 73iT^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \)
89 \( 1 + 1.91T + 89T^{2} \)
97 \( 1 + (1.07 + 1.07i)T + 97iT^{2} \)
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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

−9.320905107553482918240788201697, −8.572507224000803598083393934511, −8.011083947156145661268041620217, −7.25549481672456806461622537437, −6.45539222164589318799943643775, −5.80604338815090486150957493437, −5.03432870952641909160055767346, −3.66703119860331995541449052240, −1.87518849352011126968514577191, −0.68394552249702227674952528063, 0.837672529370471658302125839383, 2.03437575143797311971398810104, 2.92176399707531079316988019783, 3.83558349062631714091230597456, 4.79788744259671633413973200047, 6.53272274413422607216710461526, 7.07588896807156559760956491824, 8.230946403007168761120732836542, 8.732414196266822490323223235196, 9.381262174516660616410115553702

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