Properties

Label 2-7448-1.1-c1-0-22
Degree $2$
Conductor $7448$
Sign $1$
Analytic cond. $59.4725$
Root an. cond. $7.71184$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.15·3-s + 1.74·5-s − 1.65·9-s − 5.45·11-s − 3.91·13-s − 2.01·15-s − 1.01·17-s − 19-s + 7.27·23-s − 1.96·25-s + 5.39·27-s + 3.52·29-s − 5.06·31-s + 6.31·33-s − 2.61·37-s + 4.53·39-s − 11.4·41-s − 2.55·43-s − 2.88·45-s + 0.536·47-s + 1.18·51-s − 1.33·53-s − 9.49·55-s + 1.15·57-s + 10.6·59-s − 3.90·61-s − 6.82·65-s + ⋯
L(s)  = 1  − 0.669·3-s + 0.779·5-s − 0.552·9-s − 1.64·11-s − 1.08·13-s − 0.521·15-s − 0.247·17-s − 0.229·19-s + 1.51·23-s − 0.393·25-s + 1.03·27-s + 0.654·29-s − 0.910·31-s + 1.09·33-s − 0.430·37-s + 0.726·39-s − 1.78·41-s − 0.389·43-s − 0.430·45-s + 0.0782·47-s + 0.165·51-s − 0.183·53-s − 1.28·55-s + 0.153·57-s + 1.38·59-s − 0.499·61-s − 0.845·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7448\)    =    \(2^{3} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(59.4725\)
Root analytic conductor: \(7.71184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7448} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7448,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8413273837\)
\(L(\frac12)\) \(\approx\) \(0.8413273837\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 1.15T + 3T^{2} \)
5 \( 1 - 1.74T + 5T^{2} \)
11 \( 1 + 5.45T + 11T^{2} \)
13 \( 1 + 3.91T + 13T^{2} \)
17 \( 1 + 1.01T + 17T^{2} \)
23 \( 1 - 7.27T + 23T^{2} \)
29 \( 1 - 3.52T + 29T^{2} \)
31 \( 1 + 5.06T + 31T^{2} \)
37 \( 1 + 2.61T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 + 2.55T + 43T^{2} \)
47 \( 1 - 0.536T + 47T^{2} \)
53 \( 1 + 1.33T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 + 3.90T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 - 7.41T + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 - 0.214T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 5.26T + 89T^{2} \)
97 \( 1 - 7.06T + 97T^{2} \)
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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

−7.83440097063730261430757123430, −7.10844839059501744828642881020, −6.46119329125963274777284369579, −5.64096513777019208624677220575, −5.09333716897860686050420531254, −4.82084613894603822487152673551, −3.34070685697253255449727487013, −2.62546678405970884981339632285, −1.91514517869104051945203381065, −0.44884530169119687691162962628, 0.44884530169119687691162962628, 1.91514517869104051945203381065, 2.62546678405970884981339632285, 3.34070685697253255449727487013, 4.82084613894603822487152673551, 5.09333716897860686050420531254, 5.64096513777019208624677220575, 6.46119329125963274777284369579, 7.10844839059501744828642881020, 7.83440097063730261430757123430

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