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Properties

Label 2-975-13.10-c1-0-10
Degree $2$
Conductor $975$
Sign $0.618 - 0.785i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + (1.05 − 0.611i)2-s + (−0.5 − 0.866i)3-s + (−0.251 + 0.436i)4-s + (−1.05 − 0.611i)6-s + (1.72 + 0.997i)7-s + 3.06i·8-s + (−0.499 + 0.866i)9-s + (0.0539 − 0.0311i)11-s + 0.503·12-s + (−3.33 + 1.38i)13-s + 2.44·14-s + (1.36 + 2.37i)16-s + (−2.99 + 5.17i)17-s + 1.22i·18-s + (1.14 + 0.661i)19-s + ⋯
L(s)  = 1  + (0.749 − 0.432i)2-s + (−0.288 − 0.499i)3-s + (−0.125 + 0.218i)4-s + (−0.432 − 0.249i)6-s + (0.653 + 0.377i)7-s + 1.08i·8-s + (−0.166 + 0.288i)9-s + (0.0162 − 0.00938i)11-s + 0.145·12-s + (−0.923 + 0.383i)13-s + 0.652·14-s + (0.342 + 0.592i)16-s + (−0.725 + 1.25i)17-s + 0.288i·18-s + (0.263 + 0.151i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.618 - 0.785i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.618 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54356 + 0.749477i\)
\(L(\frac12)\) \(\approx\) \(1.54356 + 0.749477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (3.33 - 1.38i)T \)
good2 \( 1 + (-1.05 + 0.611i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-1.72 - 0.997i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0539 + 0.0311i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.99 - 5.17i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.14 - 0.661i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.411 + 0.713i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.04 - 7.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.04iT - 31T^{2} \)
37 \( 1 + (4.71 - 2.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.94 + 5.73i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.34 + 7.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.45iT - 47T^{2} \)
53 \( 1 - 1.39T + 53T^{2} \)
59 \( 1 + (-3.05 - 1.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.83 - 6.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.95 + 5.17i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.394 + 0.227i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.99iT - 73T^{2} \)
79 \( 1 - 8.01T + 79T^{2} \)
83 \( 1 - 4.14iT - 83T^{2} \)
89 \( 1 + (-9.96 + 5.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.2 + 5.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.53547160962344049797704137460, −9.094021158876245738507986720179, −8.459829459362056304832564159753, −7.61937692770910223058073502778, −6.66665424985270091487298031938, −5.55034180587080378143043297086, −4.84997347402989157043858273478, −3.95118230268027726796819153599, −2.66954198747060210193882246237, −1.73942582622493337755126729926, 0.65062485886227697489541787435, 2.59763095559549993672790623981, 4.02334095831573716782867378629, 4.72133211631427227136765437060, 5.31218465027213492837420929620, 6.29918329465502986433854569617, 7.19731600348225668506052312376, 8.023191412339723776739710073775, 9.399050759196829604945585324974, 9.739427703819539551663424055248

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