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Properties

Label 2-959-7.2-c1-0-0
Degree $2$
Conductor $959$
Sign $-0.906 - 0.422i$
Analytic cond. $7.65765$
Root an. cond. $2.76724$
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  + (−0.964 + 1.67i)2-s + (−1.14 − 1.98i)3-s + (−0.860 − 1.48i)4-s + (0.545 − 0.945i)5-s + 4.43·6-s + (−1.37 − 2.25i)7-s − 0.539·8-s + (−1.13 + 1.97i)9-s + (1.05 + 1.82i)10-s + (0.0464 + 0.0804i)11-s + (−1.97 + 3.42i)12-s − 5.52·13-s + (5.10 − 0.120i)14-s − 2.50·15-s + (2.24 − 3.88i)16-s + (−0.740 − 1.28i)17-s + ⋯
L(s)  = 1  + (−0.681 + 1.18i)2-s + (−0.663 − 1.14i)3-s + (−0.430 − 0.744i)4-s + (0.244 − 0.422i)5-s + 1.80·6-s + (−0.520 − 0.854i)7-s − 0.190·8-s + (−0.379 + 0.657i)9-s + (0.332 + 0.576i)10-s + (0.0140 + 0.0242i)11-s + (−0.570 + 0.988i)12-s − 1.53·13-s + (1.36 − 0.0320i)14-s − 0.647·15-s + (0.560 − 0.970i)16-s + (−0.179 − 0.311i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 959 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(959\)    =    \(7 \cdot 137\)
Sign: $-0.906 - 0.422i$
Analytic conductor: \(7.65765\)
Root analytic conductor: \(2.76724\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{959} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 959,\ (\ :1/2),\ -0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0133539 + 0.0601832i\)
\(L(\frac12)\) \(\approx\) \(0.0133539 + 0.0601832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.37 + 2.25i)T \)
137 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.964 - 1.67i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.14 + 1.98i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.545 + 0.945i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.0464 - 0.0804i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.52T + 13T^{2} \)
17 \( 1 + (0.740 + 1.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.381 - 0.660i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.68 + 4.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.44T + 29T^{2} \)
31 \( 1 + (-3.59 - 6.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.68 - 2.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.80T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + (4.90 - 8.48i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.202 - 0.351i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.15 - 8.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.38 + 7.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.763 + 1.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.98T + 71T^{2} \)
73 \( 1 + (5.16 + 8.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.83 - 8.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + (5.28 - 9.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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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

−10.08826478406109841881053263559, −9.413181552649284655033904189665, −8.482416775774076859110876596643, −7.50952721178582444319766905794, −7.03392084125939857930914690342, −6.54091344762045542274889074576, −5.56241729276447942591383064312, −4.65916226997645825488609339749, −2.84560047184828816464686748002, −1.12555166459494843945856641697, 0.04357031476461493470328929458, 2.16529448080021728003863165504, 2.96876935800024751473923676368, 4.10803314326802958272348706987, 5.25579222953188451358424924462, 5.97387237680008033169204700668, 7.12531289554478308994927732390, 8.508664958826674538738801996997, 9.350765078839532015832127545682, 9.922921544081573255608692547993

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