↑

Properties

Label 2-836-11.4-c1-0-14
Degree $2$
Conductor $836$
Sign $0.530 + 0.847i$
Analytic cond. $6.67549$
Root an. cond. $2.58369$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.145 + 0.448i)3-s + (2.86 − 2.07i)5-s + (−0.710 − 2.18i)7-s + (2.24 + 1.63i)9-s + (0.309 − 3.30i)11-s + (−0.616 − 0.448i)13-s + (0.515 + 1.58i)15-s + (−0.815 + 0.592i)17-s + (0.309 − 0.951i)19-s + 1.08·21-s − 2.94·23-s + (2.32 − 7.14i)25-s + (−2.20 + 1.60i)27-s + (−1.55 − 4.79i)29-s + (−2.67 − 1.94i)31-s + ⋯
L(s)  = 1  + (−0.0840 + 0.258i)3-s + (1.27 − 0.929i)5-s + (−0.268 − 0.826i)7-s + (0.749 + 0.544i)9-s + (0.0931 − 0.995i)11-s + (−0.171 − 0.124i)13-s + (0.132 + 0.409i)15-s + (−0.197 + 0.143i)17-s + (0.0708 − 0.218i)19-s + 0.236·21-s − 0.615·23-s + (0.464 − 1.42i)25-s + (−0.423 + 0.307i)27-s + (−0.289 − 0.890i)29-s + (−0.480 − 0.348i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 836 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(836\)    =    \(2^{2} \cdot 11 \cdot 19\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(6.67549\)
Root analytic conductor: \(2.58369\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{836} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 836,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61962 - 0.897478i\)
\(L(\frac12)\) \(\approx\) \(1.61962 - 0.897478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.309 + 3.30i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good3 \( 1 + (0.145 - 0.448i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.86 + 2.07i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.710 + 2.18i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.616 + 0.448i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.815 - 0.592i)T + (5.25 - 16.1i)T^{2} \)
23 \( 1 + 2.94T + 23T^{2} \)
29 \( 1 + (1.55 + 4.79i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.67 + 1.94i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.40 - 7.39i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.418 - 1.28i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.91T + 43T^{2} \)
47 \( 1 + (-0.146 + 0.451i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.35 - 3.16i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.583 + 1.79i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.42 + 1.75i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + (4.73 - 3.43i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.49 - 7.67i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.40 + 6.83i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.8 + 7.84i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (-2.58 - 1.87i)T + (29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00327526890731176243654193948, −9.428799009269080287185311188413, −8.488975636934770294176051067922, −7.56333609995317408582364701891, −6.42417376653456100918271491364, −5.63959228145416225990540902570, −4.74506932855823632096807300775, −3.81452835039509228310541326748, −2.22685213673611190361083605786, −0.970764598044019689316380211212, 1.77559611553381443079806086878, 2.53091632548320361124809756900, 3.87918273010408139749016658491, 5.27065439223754216331396189309, 6.09703667410294358478922755817, 6.82428051459903991430194642209, 7.47406230762899728226728069640, 8.992274293320837867874738830446, 9.572056214582184544784817357363, 10.16890951115427515339024778172

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