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Properties

Label 2-756-21.11-c2-0-5
Degree $2$
Conductor $756$
Sign $-0.311 - 0.950i$
Analytic cond. $20.5995$
Root an. cond. $4.53866$
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  + (5.5 + 4.33i)7-s − 22·13-s + (18.5 + 32.0i)19-s + (−12.5 + 21.6i)25-s + (6.5 − 11.2i)31-s + (−13 − 22.5i)37-s − 61·43-s + (11.4 + 47.6i)49-s + (60.5 + 104. i)61-s + (−61 + 105. i)67-s + (−71.5 + 123. i)73-s + (71 + 122. i)79-s + (−121 − 95.2i)91-s + 167·97-s + (−97 − 168. i)103-s + ⋯
L(s)  = 1  + (0.785 + 0.618i)7-s − 1.69·13-s + (0.973 + 1.68i)19-s + (−0.5 + 0.866i)25-s + (0.209 − 0.363i)31-s + (−0.351 − 0.608i)37-s − 1.41·43-s + (0.234 + 0.972i)49-s + (0.991 + 1.71i)61-s + (−0.910 + 1.57i)67-s + (−0.979 + 1.69i)73-s + (0.898 + 1.55i)79-s + (−1.32 − 1.04i)91-s + 1.72·97-s + (−0.941 − 1.63i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.311 - 0.950i$
Analytic conductor: \(20.5995\)
Root analytic conductor: \(4.53866\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1),\ -0.311 - 0.950i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.320505114\)
\(L(\frac12)\) \(\approx\) \(1.320505114\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-5.5 - 4.33i)T \)
good5 \( 1 + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (60.5 + 104. i)T^{2} \)
13 \( 1 + 22T + 169T^{2} \)
17 \( 1 + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-18.5 - 32.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (264.5 - 458. i)T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + (-6.5 + 11.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (13 + 22.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 61T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-60.5 - 104. i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (61 - 105. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (71.5 - 123. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-71 - 122. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 167T + 9.40e3T^{2} \)
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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

−10.18180398838282078440602391957, −9.695009576894335720349513274368, −8.637015580607987953422656196747, −7.77675929563331017016381177584, −7.13858293028610247451920256356, −5.69940630553941643181161136372, −5.19138990918447882201297068147, −4.00933821217462990339506873754, −2.68019878585152872173924439207, −1.56330464382459191433941866165, 0.44599787284632263654718859506, 2.02133196854073560470015935113, 3.22112682922620523626041015282, 4.74309213082229478332193847537, 4.98700571114432381620092994309, 6.54983830522572604608383475915, 7.35554056574911177869694079634, 8.007935522325443373220525499277, 9.116348041821956451485137785688, 9.926258634754627123060549696431

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