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Properties

Label 2-76-76.75-c7-0-10
Degree $2$
Conductor $76$
Sign $-0.0758 - 0.997i$
Analytic cond. $23.7412$
Root an. cond. $4.87250$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.6 − 3.79i)2-s + 60.8·3-s + (99.1 + 80.8i)4-s − 369.·5-s + (−648. − 230. i)6-s − 900. i·7-s + (−750. − 1.23e3i)8-s + 1.51e3·9-s + (3.93e3 + 1.40e3i)10-s + 3.23e3i·11-s + (6.03e3 + 4.92e3i)12-s + 5.36e3i·13-s + (−3.41e3 + 9.59e3i)14-s − 2.24e4·15-s + (3.29e3 + 1.60e4i)16-s + 2.36e4·17-s + ⋯
L(s)  = 1  + (−0.942 − 0.335i)2-s + 1.30·3-s + (0.774 + 0.631i)4-s − 1.32·5-s + (−1.22 − 0.436i)6-s − 0.992i·7-s + (−0.518 − 0.855i)8-s + 0.692·9-s + (1.24 + 0.443i)10-s + 0.732i·11-s + (1.00 + 0.822i)12-s + 0.677i·13-s + (−0.332 + 0.934i)14-s − 1.71·15-s + (0.201 + 0.979i)16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0758 - 0.997i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0758 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.0758 - 0.997i$
Analytic conductor: \(23.7412\)
Root analytic conductor: \(4.87250\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :7/2),\ -0.0758 - 0.997i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.532062 + 0.574058i\)
\(L(\frac12)\) \(\approx\) \(0.532062 + 0.574058i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (10.6 + 3.79i)T \)
19 \( 1 + (2.05e4 + 2.16e4i)T \)
good3 \( 1 - 60.8T + 2.18e3T^{2} \)
5 \( 1 + 369.T + 7.81e4T^{2} \)
7 \( 1 + 900. iT - 8.23e5T^{2} \)
11 \( 1 - 3.23e3iT - 1.94e7T^{2} \)
13 \( 1 - 5.36e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.36e4T + 4.10e8T^{2} \)
23 \( 1 - 1.05e5iT - 3.40e9T^{2} \)
29 \( 1 - 1.20e5iT - 1.72e10T^{2} \)
31 \( 1 + 2.25e5T + 2.75e10T^{2} \)
37 \( 1 - 4.10e5iT - 9.49e10T^{2} \)
41 \( 1 - 1.58e5iT - 1.94e11T^{2} \)
43 \( 1 + 4.24e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.01e6iT - 5.06e11T^{2} \)
53 \( 1 - 9.28e5iT - 1.17e12T^{2} \)
59 \( 1 - 3.78e5T + 2.48e12T^{2} \)
61 \( 1 - 1.27e6T + 3.14e12T^{2} \)
67 \( 1 + 6.30e5T + 6.06e12T^{2} \)
71 \( 1 + 3.25e6T + 9.09e12T^{2} \)
73 \( 1 + 4.59e6T + 1.10e13T^{2} \)
79 \( 1 - 2.29e6T + 1.92e13T^{2} \)
83 \( 1 + 3.91e6iT - 2.71e13T^{2} \)
89 \( 1 - 7.89e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.92e6iT - 8.07e13T^{2} \)
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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

−13.30960491543382956069128460591, −12.06226066483992986542498507000, −11.03857425192552010188090892696, −9.765629853056830199881273798261, −8.774041568939010446752303842899, −7.60479255635766801300834586609, −7.26789418267237289761080349241, −4.06222872937605217135989476036, −3.19635349086603137453799843147, −1.46978442228107630258809404765, 0.30850424323362801046600478231, 2.35378765154455449164111034281, 3.56847266262166874957926512517, 5.79613179110069510162893935959, 7.55774096345578200583457098506, 8.320197125020950842961845229078, 8.845545688764154299706025629006, 10.29432170217809141588384078121, 11.58213573264151265873908593568, 12.63224638757409085437099077474

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