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Properties

Label 2-968-88.69-c1-0-58
Degree $2$
Conductor $968$
Sign $0.707 + 0.707i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
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  + (−1.37 + 0.334i)2-s + (−1.22 + 0.397i)3-s + (1.77 − 0.918i)4-s + (0.929 − 1.27i)5-s + (1.54 − 0.955i)6-s + (0.577 − 1.77i)7-s + (−2.13 + 1.85i)8-s + (−1.08 + 0.790i)9-s + (−0.849 + 2.06i)10-s + (−1.80 + 1.83i)12-s + (−0.189 − 0.260i)13-s + (−0.199 + 2.63i)14-s + (−0.628 + 1.93i)15-s + (2.31 − 3.26i)16-s + (3.87 + 2.81i)17-s + (1.23 − 1.45i)18-s + ⋯
L(s)  = 1  + (−0.971 + 0.236i)2-s + (−0.706 + 0.229i)3-s + (0.888 − 0.459i)4-s + (0.415 − 0.572i)5-s + (0.632 − 0.389i)6-s + (0.218 − 0.671i)7-s + (−0.754 + 0.656i)8-s + (−0.362 + 0.263i)9-s + (−0.268 + 0.654i)10-s + (−0.521 + 0.528i)12-s + (−0.0525 − 0.0722i)13-s + (−0.0532 + 0.704i)14-s + (−0.162 + 0.499i)15-s + (0.577 − 0.816i)16-s + (0.940 + 0.683i)17-s + (0.290 − 0.341i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721824 - 0.298950i\)
\(L(\frac12)\) \(\approx\) \(0.721824 - 0.298950i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.334i)T \)
11 \( 1 \)
good3 \( 1 + (1.22 - 0.397i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.929 + 1.27i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.577 + 1.77i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.189 + 0.260i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.87 - 2.81i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.22 + 1.04i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.08T + 23T^{2} \)
29 \( 1 + (9.90 + 3.21i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.28 + 1.65i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.39 + 1.10i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.39 - 4.27i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.61iT - 43T^{2} \)
47 \( 1 + (3.93 + 12.1i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.32 + 4.57i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-5.12 - 1.66i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.42 + 8.84i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + (-9.00 - 6.54i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.89 + 11.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.83 + 3.51i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.0820 - 0.112i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.28T + 89T^{2} \)
97 \( 1 + (10.0 - 7.28i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.876003679766267245231136092605, −9.211162852983583356613050615019, −8.205728352752561187588838876238, −7.55194283751180264695803021388, −6.55023489300752846213743099807, −5.51611791608377266904299253926, −5.12415627827339486400336496780, −3.51381552446040996960325529039, −1.90818653350507229360279560584, −0.65102578199190627251210137489, 1.12025577800242444067378642598, 2.51722515151902376779830726024, 3.39120656296870578688432184577, 5.28824531716374325598090559044, 5.92376619987178076380689537366, 6.84973891097074301313989796758, 7.54259426575921342946728438833, 8.615955098697171781244421712451, 9.365393811473010950977652436505, 10.06761948422556487933971467504

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