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Properties

Label 2-768-96.35-c1-0-6
Degree $2$
Conductor $768$
Sign $-0.609 - 0.792i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
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  + (−0.0499 + 1.73i)3-s + (1.06 − 0.443i)5-s + (2.37 + 2.37i)7-s + (−2.99 − 0.173i)9-s + (−5.50 + 2.27i)11-s + (−0.346 + 0.836i)13-s + (0.713 + 1.87i)15-s − 0.685·17-s + (3.35 + 1.38i)19-s + (−4.22 + 3.98i)21-s + (2.05 + 2.05i)23-s + (−2.58 + 2.58i)25-s + (0.449 − 5.17i)27-s + (1.98 − 4.80i)29-s + 6.36i·31-s + ⋯
L(s)  = 1  + (−0.0288 + 0.999i)3-s + (0.478 − 0.198i)5-s + (0.896 + 0.896i)7-s + (−0.998 − 0.0576i)9-s + (−1.65 + 0.687i)11-s + (−0.0960 + 0.231i)13-s + (0.184 + 0.483i)15-s − 0.166·17-s + (0.768 + 0.318i)19-s + (−0.922 + 0.870i)21-s + (0.427 + 0.427i)23-s + (−0.517 + 0.517i)25-s + (0.0864 − 0.996i)27-s + (0.369 − 0.891i)29-s + 1.14i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.609 - 0.792i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ -0.609 - 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.613165 + 1.24438i\)
\(L(\frac12)\) \(\approx\) \(0.613165 + 1.24438i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.0499 - 1.73i)T \)
good5 \( 1 + (-1.06 + 0.443i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.37 - 2.37i)T + 7iT^{2} \)
11 \( 1 + (5.50 - 2.27i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.346 - 0.836i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 0.685T + 17T^{2} \)
19 \( 1 + (-3.35 - 1.38i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.05 - 2.05i)T + 23iT^{2} \)
29 \( 1 + (-1.98 + 4.80i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.36iT - 31T^{2} \)
37 \( 1 + (0.112 + 0.272i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.26 - 3.26i)T - 41iT^{2} \)
43 \( 1 + (-0.993 - 2.39i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 + (-1.56 - 3.77i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.20 - 5.33i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.53 - 0.636i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (4.69 - 11.3i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (7.99 - 7.99i)T - 71iT^{2} \)
73 \( 1 + (2.34 + 2.34i)T + 73iT^{2} \)
79 \( 1 - 8.91T + 79T^{2} \)
83 \( 1 + (-3.06 + 7.40i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (1.99 + 1.99i)T + 89iT^{2} \)
97 \( 1 - 8.66T + 97T^{2} \)
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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

−10.37682090231842040355857313019, −9.919104305576540245361939459500, −8.959448111331661006830473953312, −8.268546922900439927775832577951, −7.33098916060238739266889277129, −5.72618408913190203432487817377, −5.27016797076082183470193755341, −4.53985372965560239688628550137, −3.03264082364889581448680697860, −2.00842595944089764855027723353, 0.68533778112922212669424213277, 2.13104368947734317000090317138, 3.14552892107593679734899768661, 4.81336054579163645202547309207, 5.59585784648166416291850174167, 6.58548498653633210997535860139, 7.71076689869597837218918233261, 7.88335052944500942979796803876, 9.007379113448834893432813888028, 10.30838185173267719477334002093

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