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Properties

Label 2-76-76.11-c2-0-16
Degree $2$
Conductor $76$
Sign $0.327 + 0.944i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + (−4.23 − 2.44i)3-s + 4·4-s + (1.73 − 3.01i)5-s + (−8.47 − 4.89i)6-s − 10.0i·7-s + 8·8-s + (7.47 + 12.9i)9-s + (3.47 − 6.02i)10-s + 17.5i·11-s + (−16.9 − 9.78i)12-s + (−3 − 5.19i)13-s − 20.1i·14-s + (−14.7 + 8.50i)15-s + 16·16-s + (−2.52 + 4.36i)17-s + ⋯
L(s)  = 1  + 2-s + (−1.41 − 0.815i)3-s + 4-s + (0.347 − 0.602i)5-s + (−1.41 − 0.815i)6-s − 1.44i·7-s + 8-s + (0.830 + 1.43i)9-s + (0.347 − 0.602i)10-s + 1.59i·11-s + (−1.41 − 0.815i)12-s + (−0.230 − 0.399i)13-s − 1.44i·14-s + (−0.982 + 0.567i)15-s + 16-s + (−0.148 + 0.257i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.327 + 0.944i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.327 + 0.944i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.27254 - 0.906015i\)
\(L(\frac12)\) \(\approx\) \(1.27254 - 0.906015i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
19 \( 1 + (-2.28 - 18.8i)T \)
good3 \( 1 + (4.23 + 2.44i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.73 + 3.01i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + 10.0iT - 49T^{2} \)
11 \( 1 - 17.5iT - 121T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (2.52 - 4.36i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (-27.6 + 15.9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (3.73 + 6.47i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 4.81iT - 961T^{2} \)
37 \( 1 + 51.3T + 1.36e3T^{2} \)
41 \( 1 + (-17.4 + 30.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-9.38 - 5.41i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (19.1 - 11.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-20.0 - 34.7i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (16.7 + 9.67i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (8.26 + 14.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (83.0 - 47.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-12.6 - 7.30i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (17.4 - 30.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (68.0 + 39.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 92.9iT - 6.88e3T^{2} \)
89 \( 1 + (-56.2 - 97.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (63.9 - 110. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.66519706695561773624142028331, −12.71939072009457375750230232268, −12.34043629979327571778024230806, −10.95768244825309701870229730382, −10.16613175371278255627610311478, −7.46584176149451481536715693669, −6.82585109614242021533208711003, −5.42796947931396760924701382771, −4.41314192393084445856668560605, −1.42021795205647615767728878716, 3.03558867566400386420852005128, 4.98151321185819561980978028228, 5.77623636050006051218960637184, 6.67959576744378280621532437844, 9.049295005627654440849715377758, 10.59511886577952237863530976342, 11.35751416752391364470686178987, 11.97981147583173293340497828141, 13.35111892061791544503568816264, 14.62947849025991802165455290625

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