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Properties

Label 2-1575-105.89-c1-0-25
Degree $2$
Conductor $1575$
Sign $0.472 - 0.881i$
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  + (−0.707 + 1.22i)2-s + (2.59 − 0.5i)7-s − 2.82·8-s + (−1.22 + 0.707i)11-s + 5.19·13-s + (−1.22 + 3.53i)14-s + (2.00 − 3.46i)16-s + (4.24 − 2.44i)17-s + (−1.5 − 0.866i)19-s − 2i·22-s + (2.82 − 4.89i)23-s + (−3.67 + 6.36i)26-s − 2.82i·29-s + (1.5 − 0.866i)31-s + 6.92i·34-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)2-s + (0.981 − 0.188i)7-s − 0.999·8-s + (−0.369 + 0.213i)11-s + 1.44·13-s + (−0.327 + 0.944i)14-s + (0.500 − 0.866i)16-s + (1.02 − 0.594i)17-s + (−0.344 − 0.198i)19-s − 0.426i·22-s + (0.589 − 1.02i)23-s + (−0.720 + 1.24i)26-s − 0.525i·29-s + (0.269 − 0.155i)31-s + 1.18i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.472 - 0.881i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.472 - 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.580926429\)
\(L(\frac12)\) \(\approx\) \(1.580926429\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 0.5i)T \)
good2 \( 1 + (0.707 - 1.22i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (1.22 - 0.707i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.19T + 13T^{2} \)
17 \( 1 + (-4.24 + 2.44i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.82 + 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.34T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + (-10.6 - 6.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.41 + 2.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.44 - 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.52 + 5.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + (0.866 + 1.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.34iT - 83T^{2} \)
89 \( 1 + (2.44 - 4.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−9.251991058941808613873164525253, −8.508319490713306062594542029807, −8.019216685765678739198102745719, −7.30071923348259506518796409294, −6.46207008553003048725581399676, −5.66675715003781430367312576352, −4.74499301895655256745254747944, −3.62541384226967002448581676508, −2.49741836211945352553979329941, −0.954812901784168572942617613920, 1.09044688085996643423897744174, 1.82861426733314264431994659458, 3.09190002670332119830228443292, 3.92157843483627024840577532895, 5.35724117473579178668648792409, 5.80989218522265658310514084360, 6.92042551085315255485005264585, 8.098155422474624095402739022145, 8.540918530739692847497832861665, 9.327673944293292399160380335922

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