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Properties

Label 2-3104-776.35-c0-0-0
Degree $2$
Conductor $3104$
Sign $-0.421 - 0.907i$
Analytic cond. $1.54909$
Root an. cond. $1.24462$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)11-s + (−1 + 1.73i)17-s − 2·19-s + (−0.5 + 0.866i)25-s − 27-s + 0.999·33-s + (−1 − 1.73i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)49-s + 1.99·51-s + (1 + 1.73i)57-s + (1 + 1.73i)59-s + 67-s + (−1 − 1.73i)73-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)11-s + (−1 + 1.73i)17-s − 2·19-s + (−0.5 + 0.866i)25-s − 27-s + 0.999·33-s + (−1 − 1.73i)41-s + (−0.5 − 0.866i)43-s + (−0.5 + 0.866i)49-s + 1.99·51-s + (1 + 1.73i)57-s + (1 + 1.73i)59-s + 67-s + (−1 − 1.73i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3104\)    =    \(2^{5} \cdot 97\)
Sign: $-0.421 - 0.907i$
Analytic conductor: \(1.54909\)
Root analytic conductor: \(1.24462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3104} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3104,\ (\ :0),\ -0.421 - 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2450395040\)
\(L(\frac12)\) \(\approx\) \(0.2450395040\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
97 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + 2T + T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
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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

−8.879787274606681813387442039881, −8.392242314109835285515641949561, −7.44059513242000742027632490418, −6.84034837741633518865792693516, −6.22481607868057424420343733451, −5.51445091251453794884501302710, −4.39156294253239848070930361960, −3.77528068949820579157003501416, −2.18803829276748699923283381460, −1.70824942684339872176437601431, 0.14588487962323042740174787256, 2.09794552152422887479601829294, 3.02689993280511953731454274230, 4.18535654232123456682614174734, 4.72011249489251468241012098800, 5.42957215311863395141389801652, 6.36238595225842379503116449729, 6.94693610498858726653946230003, 8.211157577544965359464925685891, 8.480831576671176618475018414355

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