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Properties

Label 2-2e7-32.5-c1-0-2
Degree $2$
Conductor $128$
Sign $-0.555 + 0.831i$
Analytic cond. $1.02208$
Root an. cond. $1.01098$
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  + (−0.707 − 1.70i)3-s + (−3.12 − 1.29i)5-s + (−1 − i)7-s + (−0.292 + 0.292i)9-s + (−0.121 + 0.292i)11-s + (1.70 − 0.707i)13-s + 6.24i·15-s − 2.82i·17-s + (5.53 − 2.29i)19-s + (−0.999 + 2.41i)21-s + (−0.171 + 0.171i)23-s + (4.53 + 4.53i)25-s + (−4.41 − 1.82i)27-s + (1.12 + 2.70i)29-s + 4·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.985i)3-s + (−1.39 − 0.578i)5-s + (−0.377 − 0.377i)7-s + (−0.0976 + 0.0976i)9-s + (−0.0365 + 0.0883i)11-s + (0.473 − 0.196i)13-s + 1.61i·15-s − 0.685i·17-s + (1.26 − 0.526i)19-s + (−0.218 + 0.526i)21-s + (−0.0357 + 0.0357i)23-s + (0.907 + 0.907i)25-s + (−0.849 − 0.351i)27-s + (0.208 + 0.502i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(1.02208\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ -0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321577 - 0.601628i\)
\(L(\frac12)\) \(\approx\) \(0.321577 - 0.601628i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.707 + 1.70i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (3.12 + 1.29i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (0.121 - 0.292i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.70 + 0.707i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 2.82iT - 17T^{2} \)
19 \( 1 + (-5.53 + 2.29i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.171 - 0.171i)T - 23iT^{2} \)
29 \( 1 + (-1.12 - 2.70i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-1.70 - 0.707i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.82 - 5.82i)T - 41iT^{2} \)
43 \( 1 + (3.29 - 7.94i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + (-3.12 + 7.53i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-6.12 - 2.53i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.292 - 0.707i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (1.53 + 3.70i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-0.171 - 0.171i)T + 71iT^{2} \)
73 \( 1 + (-7 + 7i)T - 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (6.12 - 2.53i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (2.65 + 2.65i)T + 89iT^{2} \)
97 \( 1 + 1.51T + 97T^{2} \)
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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

−12.93263907815785544907306799070, −11.85220318953463728726891367140, −11.50008300591082066857360547155, −9.866627859539887801784950182392, −8.440810548736777930964437681791, −7.48059168041753314794545315847, −6.66741511244973805435690844446, −4.98923570839777185369827673984, −3.50359414138225868731632229317, −0.802755396945065606216234866833, 3.36055108308096228739991492130, 4.32411170575179470425189897881, 5.80751545761279233116925922879, 7.26802842249833095936075354794, 8.384119826253477384487204040967, 9.743432227012172835864545995063, 10.70152564133186421694923558469, 11.51821805194461738549858974121, 12.34438350287135920762462750338, 13.82294105993128630645716981317

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