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Properties

Label 2-952-952.781-c1-0-94
Degree $2$
Conductor $952$
Sign $-0.393 + 0.919i$
Analytic cond. $7.60175$
Root an. cond. $2.75712$
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  + (−1.41 − 0.0360i)2-s + (0.193 − 0.335i)3-s + (1.99 + 0.102i)4-s + (−0.731 − 1.26i)5-s + (−0.285 + 0.466i)6-s + (0.144 − 2.64i)7-s + (−2.82 − 0.216i)8-s + (1.42 + 2.46i)9-s + (0.988 + 1.81i)10-s + (−3.18 + 5.51i)11-s + (0.420 − 0.649i)12-s − 3.00i·13-s + (−0.299 + 3.72i)14-s − 0.566·15-s + (3.97 + 0.407i)16-s + (0.857 − 4.03i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0255i)2-s + (0.111 − 0.193i)3-s + (0.998 + 0.0510i)4-s + (−0.327 − 0.566i)5-s + (−0.116 + 0.190i)6-s + (0.0546 − 0.998i)7-s + (−0.997 − 0.0764i)8-s + (0.475 + 0.822i)9-s + (0.312 + 0.574i)10-s + (−0.960 + 1.66i)11-s + (0.121 − 0.187i)12-s − 0.834i·13-s + (−0.0801 + 0.996i)14-s − 0.146·15-s + (0.994 + 0.101i)16-s + (0.208 − 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(952\)    =    \(2^{3} \cdot 7 \cdot 17\)
Sign: $-0.393 + 0.919i$
Analytic conductor: \(7.60175\)
Root analytic conductor: \(2.75712\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{952} (781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 952,\ (\ :1/2),\ -0.393 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420604 - 0.637403i\)
\(L(\frac12)\) \(\approx\) \(0.420604 - 0.637403i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0360i)T \)
7 \( 1 + (-0.144 + 2.64i)T \)
17 \( 1 + (-0.857 + 4.03i)T \)
good3 \( 1 + (-0.193 + 0.335i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.731 + 1.26i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.18 - 5.51i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.00iT - 13T^{2} \)
19 \( 1 + (-4.14 + 2.39i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.48 + 2.58i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.40T + 29T^{2} \)
31 \( 1 + (3.36 + 1.94i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.698 + 1.20i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.18iT - 41T^{2} \)
43 \( 1 - 7.01iT - 43T^{2} \)
47 \( 1 + (4.75 + 8.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.221 + 0.127i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.60 - 2.08i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.37 + 5.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.1 + 5.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.81iT - 71T^{2} \)
73 \( 1 + (13.4 + 7.75i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.17 + 4.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.2iT - 83T^{2} \)
89 \( 1 + (-5.98 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.64iT - 97T^{2} \)
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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

−9.866212334824397006966537198541, −8.992160363579642589284157058598, −7.77158889841965338259837917343, −7.54241506877627209075936862635, −6.95512594180115112416052518772, −5.25278503814120502183951985992, −4.61955217204084965672652963354, −3.04605454941916388253978342355, −1.86923533804243929530390566955, −0.50369967301979049563900945400, 1.44401921335999113662359242078, 2.98923495072532451500129758864, 3.52886737182374438811623488732, 5.45666338718583612363126181691, 6.08943347988479317688685456850, 7.09605249439789886588405852425, 7.913214232749951007319330839015, 8.801549690402072297588669713086, 9.275072615392346256304640820819, 10.22915338864288753546703135838

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