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Properties

Label 2-3104-776.99-c0-0-0
Degree $2$
Conductor $3104$
Sign $0.308 - 0.951i$
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  + (0.207 − 0.158i)3-s + (−0.241 + 0.900i)9-s + (−0.0675 − 0.513i)11-s + (−0.128 + 1.95i)17-s + (−0.923 + 1.38i)19-s + (−0.608 + 0.793i)25-s + (0.192 + 0.465i)27-s + (−0.0955 − 0.0955i)33-s + (1.05 + 0.357i)41-s + (−0.410 − 1.53i)43-s + (0.991 − 0.130i)49-s + (0.284 + 0.425i)51-s + (0.0283 + 0.433i)57-s + (0.257 − 0.293i)59-s + (1.57 + 1.05i)67-s + ⋯
L(s)  = 1  + (0.207 − 0.158i)3-s + (−0.241 + 0.900i)9-s + (−0.0675 − 0.513i)11-s + (−0.128 + 1.95i)17-s + (−0.923 + 1.38i)19-s + (−0.608 + 0.793i)25-s + (0.192 + 0.465i)27-s + (−0.0955 − 0.0955i)33-s + (1.05 + 0.357i)41-s + (−0.410 − 1.53i)43-s + (0.991 − 0.130i)49-s + (0.284 + 0.425i)51-s + (0.0283 + 0.433i)57-s + (0.257 − 0.293i)59-s + (1.57 + 1.05i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3104\)    =    \(2^{5} \cdot 97\)
Sign: $0.308 - 0.951i$
Analytic conductor: \(1.54909\)
Root analytic conductor: \(1.24462\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3104} (1263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3104,\ (\ :0),\ 0.308 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100595983\)
\(L(\frac12)\) \(\approx\) \(1.100595983\)
\(L(1)\) 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 \( 1 + (0.793 - 0.608i)T \)
good3 \( 1 + (-0.207 + 0.158i)T + (0.258 - 0.965i)T^{2} \)
5 \( 1 + (0.608 - 0.793i)T^{2} \)
7 \( 1 + (-0.991 + 0.130i)T^{2} \)
11 \( 1 + (0.0675 + 0.513i)T + (-0.965 + 0.258i)T^{2} \)
13 \( 1 + (-0.608 + 0.793i)T^{2} \)
17 \( 1 + (0.128 - 1.95i)T + (-0.991 - 0.130i)T^{2} \)
19 \( 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2} \)
23 \( 1 + (0.130 - 0.991i)T^{2} \)
29 \( 1 + (0.793 + 0.608i)T^{2} \)
31 \( 1 + (0.258 - 0.965i)T^{2} \)
37 \( 1 + (-0.130 - 0.991i)T^{2} \)
41 \( 1 + (-1.05 - 0.357i)T + (0.793 + 0.608i)T^{2} \)
43 \( 1 + (0.410 + 1.53i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.965 + 0.258i)T^{2} \)
59 \( 1 + (-0.257 + 0.293i)T + (-0.130 - 0.991i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.57 - 1.05i)T + (0.382 + 0.923i)T^{2} \)
71 \( 1 + (-0.793 + 0.608i)T^{2} \)
73 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (1.50 + 0.0983i)T + (0.991 + 0.130i)T^{2} \)
89 \( 1 + (1.12 + 0.465i)T + (0.707 + 0.707i)T^{2} \)
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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

−8.736347362350989212313062800588, −8.284818384137713390637196575219, −7.75966656881997772296137487313, −6.76737189081060869401591275106, −5.86974362211968950700419100202, −5.44501787359434701888215029539, −4.14588438121690093189640279078, −3.62754927260318928381352669345, −2.34052816441802697697617696023, −1.59984221475089465836789092206, 0.64576724806231953065861426128, 2.34925109558118133221651070436, 2.94914985468286081625294431608, 4.14369019923536391000171761255, 4.72232347847113313240748201334, 5.67971229344761344918188463177, 6.65715725576688099690521645807, 7.07588681225569484351274685595, 8.051014113988557240143178145862, 8.841132662396466021991819439331

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