Properties

Label 2-1216-152.75-c1-0-8
Degree $2$
Conductor $1216$
Sign $-0.707 - 0.707i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.35i·5-s + 3i·7-s + 3·9-s − 4.35·11-s + 7·17-s + 4.35·19-s + 4i·23-s − 14.0·25-s − 13.0·35-s − 13.0·43-s + 13.0i·45-s − 13i·47-s − 2·49-s − 19.0i·55-s + 4.35i·61-s + ⋯
L(s)  = 1  + 1.94i·5-s + 1.13i·7-s + 9-s − 1.31·11-s + 1.69·17-s + 1.00·19-s + 0.834i·23-s − 2.80·25-s − 2.21·35-s − 1.99·43-s + 1.94i·45-s − 1.89i·47-s − 0.285·49-s − 2.56i·55-s + 0.558i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.546655545\)
\(L(\frac12)\) \(\approx\) \(1.546655545\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.35T \)
good3 \( 1 - 3T^{2} \)
5 \( 1 - 4.35iT - 5T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + 4.35T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 13.0T + 43T^{2} \)
47 \( 1 + 13iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 4.35iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.71T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.11585355743316227076640962654, −9.570762131679372410567805352438, −8.133096166120605187853632754073, −7.49634751387722474961562226909, −6.88805845467818410222614558906, −5.78458890582269204569409955706, −5.23385125515505769292448144974, −3.52967760534338542679190086557, −2.97723176560102034372782779674, −1.93680931932205370632783871178, 0.69911046023946851255477402298, 1.52586310849451429400949758104, 3.36368190604318474366024854417, 4.47050555008448355340762828076, 4.96565149590601462395531688317, 5.81263620916502455403853978296, 7.28019754241498009659040552412, 7.82853368535586807523609106040, 8.446862728355132812772884625237, 9.749393363677848593871676207406

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