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Properties

Label 2-76-76.27-c1-0-6
Degree $2$
Conductor $76$
Sign $0.140 + 0.990i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
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  + (0.112 − 1.40i)2-s + (−0.305 + 0.528i)3-s + (−1.97 − 0.315i)4-s + (1.59 − 2.75i)5-s + (0.710 + 0.489i)6-s − 2.36i·7-s + (−0.666 + 2.74i)8-s + (1.31 + 2.27i)9-s + (−3.71 − 2.55i)10-s + 5.46i·11-s + (0.769 − 0.947i)12-s + (−2.31 + 1.33i)13-s + (−3.33 − 0.264i)14-s + (0.971 + 1.68i)15-s + (3.80 + 1.24i)16-s + (−0.552 + 0.957i)17-s + ⋯
L(s)  = 1  + (0.0792 − 0.996i)2-s + (−0.176 + 0.305i)3-s + (−0.987 − 0.157i)4-s + (0.712 − 1.23i)5-s + (0.290 + 0.199i)6-s − 0.893i·7-s + (−0.235 + 0.971i)8-s + (0.437 + 0.758i)9-s + (−1.17 − 0.807i)10-s + 1.64i·11-s + (0.222 − 0.273i)12-s + (−0.643 + 0.371i)13-s + (−0.890 − 0.0707i)14-s + (0.250 + 0.434i)15-s + (0.950 + 0.312i)16-s + (−0.134 + 0.232i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.140 + 0.990i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.140 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.700177 - 0.607930i\)
\(L(\frac12)\) \(\approx\) \(0.700177 - 0.607930i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.112 + 1.40i)T \)
19 \( 1 + (-1.37 + 4.13i)T \)
good3 \( 1 + (0.305 - 0.528i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.59 + 2.75i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.36iT - 7T^{2} \)
11 \( 1 - 5.46iT - 11T^{2} \)
13 \( 1 + (2.31 - 1.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.552 - 0.957i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.46 + 1.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.63 - 3.25i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.01T + 31T^{2} \)
37 \( 1 - 0.450iT - 37T^{2} \)
41 \( 1 + (-0.336 - 0.194i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.96 + 2.86i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.91 + 1.68i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.53 - 2.03i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.82 - 11.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.77 + 11.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.27 + 7.39i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.07 + 1.86i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.91 + 6.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.57 - 9.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.14iT - 83T^{2} \)
89 \( 1 + (-4.19 + 2.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.641 + 0.370i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.81295749148981998492719576021, −13.05059032374508947110258798534, −12.28718015213445901303951044948, −10.80779837141235436704331660685, −9.863439777929335465163330647386, −9.130685297236093830457794826838, −7.40048997382826132366272954710, −5.01957485296628811237563180016, −4.47893144788155492923316633815, −1.81426115304537942798556109579, 3.24673229755392597527907531266, 5.67941136061964074012185694937, 6.30223786174490233161982069426, 7.56283604280885893432866799465, 9.017443922883007842439261129229, 10.08112512597867824656858773105, 11.58457027096739231197712364669, 12.89850789529662507894387079086, 13.95194905806316685771317776432, 14.78187435576124284091708420342

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