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Properties

Label 2-3152-788.787-c0-0-2
Degree $2$
Conductor $3152$
Sign $1$
Analytic cond. $1.57305$
Root an. cond. $1.25421$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.90·3-s + 2.61·9-s + 1.17·11-s + 25-s − 3.07·27-s − 1.61·29-s − 1.17·31-s − 2.23·33-s + 0.618·37-s + 0.618·41-s + 49-s − 0.618·53-s − 0.618·61-s + 1.90·67-s + 1.90·71-s − 1.90·75-s + 1.17·79-s + 3.23·81-s + 3.07·87-s + 2.23·93-s − 1.61·97-s + 3.07·99-s + 1.61·101-s − 1.61·109-s − 1.17·111-s + ⋯
L(s)  = 1  − 1.90·3-s + 2.61·9-s + 1.17·11-s + 25-s − 3.07·27-s − 1.61·29-s − 1.17·31-s − 2.23·33-s + 0.618·37-s + 0.618·41-s + 49-s − 0.618·53-s − 0.618·61-s + 1.90·67-s + 1.90·71-s − 1.90·75-s + 1.17·79-s + 3.23·81-s + 3.07·87-s + 2.23·93-s − 1.61·97-s + 3.07·99-s + 1.61·101-s − 1.61·109-s − 1.17·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3152\)    =    \(2^{4} \cdot 197\)
Sign: $1$
Analytic conductor: \(1.57305\)
Root analytic conductor: \(1.25421\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3152} (3151, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3152,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6977951422\)
\(L(\frac12)\) \(\approx\) \(0.6977951422\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
197 \( 1 + T \)
good3 \( 1 + 1.90T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.17T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.61T + T^{2} \)
31 \( 1 + 1.17T + T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.618T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 - 1.90T + T^{2} \)
71 \( 1 - 1.90T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.17T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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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

−9.210966083762085535941315876649, −7.88598932752584414930891824244, −7.02200854386659272174245981424, −6.59233704869861490838658886034, −5.77318186705446501231496602919, −5.23504314445205210401291093210, −4.32331499175253833503287790504, −3.65911108167422083391411103881, −1.90405204666278387202367741568, −0.852846778335477092953288929705, 0.852846778335477092953288929705, 1.90405204666278387202367741568, 3.65911108167422083391411103881, 4.32331499175253833503287790504, 5.23504314445205210401291093210, 5.77318186705446501231496602919, 6.59233704869861490838658886034, 7.02200854386659272174245981424, 7.88598932752584414930891824244, 9.210966083762085535941315876649

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