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Properties

Label 2-3724-931.645-c0-0-1
Degree $2$
Conductor $3724$
Sign $0.201 + 0.979i$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
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  + (1.32 − 0.636i)5-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)9-s + (−1.23 − 1.54i)11-s + (0.326 + 1.42i)17-s + 19-s + (−0.277 + 1.21i)23-s + (0.716 − 0.898i)25-s + (−0.535 − 1.36i)35-s + (−1.72 − 0.829i)43-s + (0.326 − 1.42i)45-s + (1.03 + 1.29i)47-s + (−0.988 − 0.149i)49-s + (−2.61 − 1.25i)55-s + (−0.162 − 0.712i)61-s + ⋯
L(s)  = 1  + (1.32 − 0.636i)5-s + (0.0747 − 0.997i)7-s + (0.623 − 0.781i)9-s + (−1.23 − 1.54i)11-s + (0.326 + 1.42i)17-s + 19-s + (−0.277 + 1.21i)23-s + (0.716 − 0.898i)25-s + (−0.535 − 1.36i)35-s + (−1.72 − 0.829i)43-s + (0.326 − 1.42i)45-s + (1.03 + 1.29i)47-s + (−0.988 − 0.149i)49-s + (−2.61 − 1.25i)55-s + (−0.162 − 0.712i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.201 + 0.979i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.201 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.634142912\)
\(L(\frac12)\) \(\approx\) \(1.634142912\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.0747 + 0.997i)T \)
19 \( 1 - T \)
good3 \( 1 + (-0.623 + 0.781i)T^{2} \)
5 \( 1 + (-1.32 + 0.636i)T + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (1.23 + 1.54i)T + (-0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.326 - 1.42i)T + (-0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (1.72 + 0.829i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.623 - 0.781i)T + (-0.222 - 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.516490405104941077574809042738, −7.88364142515167743243863163788, −7.07046723742700920630941331921, −6.03293775711696733181696974145, −5.71457436011275028205348367001, −4.89837261462315413515529273618, −3.77443228497539185730194941776, −3.16007131535767056723422404747, −1.72589580248725352387581059383, −0.975675358263967733189710571322, 1.79579290002447966101223331270, 2.39783299661336082264785445724, 2.95501376971164426508123553997, 4.68391651785852741888209582518, 5.10147001662350978140847402981, 5.73275191867270279462652496224, 6.74263961166573835731161476809, 7.31001781868437718185027616662, 7.993118158610964030592157523681, 9.021761041063392783211477696569

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