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Properties

Label 2-1575-21.17-c1-0-38
Degree $2$
Conductor $1575$
Sign $0.0285 + 0.999i$
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  + (−1.22 + 0.707i)2-s + (0.5 − 2.59i)7-s − 2.82i·8-s + (−1.22 − 0.707i)11-s + 5.19i·13-s + (1.22 + 3.53i)14-s + (2.00 + 3.46i)16-s + (2.44 − 4.24i)17-s + (1.5 − 0.866i)19-s + 2·22-s + (−4.89 + 2.82i)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  + (−0.866 + 0.499i)2-s + (0.188 − 0.981i)7-s − 0.999i·8-s + (−0.369 − 0.213i)11-s + 1.44i·13-s + (0.327 + 0.944i)14-s + (0.500 + 0.866i)16-s + (0.594 − 1.02i)17-s + (0.344 − 0.198i)19-s + 0.426·22-s + (−1.02 + 0.589i)23-s + (−0.720 − 1.24i)26-s − 0.525i·29-s + (0.269 + 0.155i)31-s + 1.18i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0285 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0285 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.0285 + 0.999i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.0285 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4908569279\)
\(L(\frac12)\) \(\approx\) \(0.4908569279\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 2.59i)T \)
good2 \( 1 + (1.22 - 0.707i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + (-2.44 + 4.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.89 - 2.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.34T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (6.12 + 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.44 + 1.41i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 + (1.5 + 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 + (-2.44 - 4.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−9.251646579606457725570463616768, −8.335119682917507795545405772491, −7.65057973062297480852951450343, −7.06032907152913407919688234812, −6.35376972859363874496400785043, −5.06127261786452702038868299782, −4.14258149937351857663980238326, −3.29581494214728492448581621492, −1.65619169685046620790635569729, −0.27587541648286738947509942211, 1.31014691201399497390169003224, 2.37687159569534293598171328943, 3.30459046626302054501277281538, 4.80890258032329613704687775350, 5.57229399138998772021144677362, 6.18722293082823443726536666062, 7.72254936078861763013555739277, 8.202072855054626173012037032922, 8.769460424719282397230800744895, 9.853041989291713805400777150602

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