↑

Properties

Label 2-1216-19.11-c1-0-17
Degree $2$
Conductor $1216$
Sign $0.899 - 0.437i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  + (−0.0731 + 0.126i)3-s + (1.17 − 2.02i)5-s + 3.83·7-s + (1.48 + 2.57i)9-s − 3.34·11-s + (3.08 + 5.34i)13-s + (0.171 + 0.296i)15-s + (−2.59 + 4.50i)17-s + (3.01 + 3.14i)19-s + (−0.280 + 0.485i)21-s + (−1.17 − 2.02i)23-s + (−0.244 − 0.423i)25-s − 0.875·27-s + (0.0250 + 0.0434i)29-s − 3.43·31-s + ⋯
L(s)  = 1  + (−0.0422 + 0.0731i)3-s + (0.523 − 0.907i)5-s + 1.44·7-s + (0.496 + 0.859i)9-s − 1.00·11-s + (0.856 + 1.48i)13-s + (0.0442 + 0.0766i)15-s + (−0.630 + 1.09i)17-s + (0.691 + 0.722i)19-s + (−0.0611 + 0.106i)21-s + (−0.244 − 0.423i)23-s + (−0.0489 − 0.0847i)25-s − 0.168·27-s + (0.00466 + 0.00807i)29-s − 0.617·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.899 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.097235413\)
\(L(\frac12)\) \(\approx\) \(2.097235413\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-3.01 - 3.14i)T \)
good3 \( 1 + (0.0731 - 0.126i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.17 + 2.02i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.83T + 7T^{2} \)
11 \( 1 + 3.34T + 11T^{2} \)
13 \( 1 + (-3.08 - 5.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.59 - 4.50i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.17 + 2.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.0250 - 0.0434i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.43T + 31T^{2} \)
37 \( 1 - 5.43T + 37T^{2} \)
41 \( 1 + (-3.64 + 6.31i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.43 - 7.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.36 + 9.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.59 + 2.76i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.92 + 3.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.46 - 2.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.75 + 9.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.744 - 1.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.84 + 6.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0875 - 0.151i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.00T + 83T^{2} \)
89 \( 1 + (-6.77 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.84 + 3.19i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.770166322590181775265640911425, −8.827231332008786329425121383696, −8.215884898629955445851482035577, −7.59593020689808970203386651789, −6.34199193711463383212115088585, −5.32330839532267357988486093995, −4.77391438896302216697147392592, −3.96996044254100825749289403132, −1.98637835839650462588138682701, −1.57728326419310450036423522362, 1.00313294311505007111857187465, 2.43627777993231955579026729507, 3.28500494723553545414247793033, 4.66083501272467713262795660406, 5.45260765109389513117600446492, 6.30842123485363692564400742133, 7.36202395331450260485383615907, 7.84726949083662395167030584972, 8.853589849623565927586448649874, 9.795457696771280657366040578448

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