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Properties

Label 2-3724-931.113-c0-0-1
Degree $2$
Conductor $3724$
Sign $0.325 + 0.945i$
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  + (1.03 − 1.29i)5-s + (0.365 + 0.930i)7-s + (−0.222 − 0.974i)9-s + (0.326 − 1.42i)11-s + (−1.48 + 0.716i)17-s + 19-s + (0.400 + 0.193i)23-s + (−0.385 − 1.68i)25-s + (1.57 + 0.487i)35-s + (0.0931 + 0.116i)43-s + (−1.48 − 0.716i)45-s + (0.440 − 1.92i)47-s + (−0.733 + 0.680i)49-s + (−1.51 − 1.89i)55-s + (−1.72 + 0.829i)61-s + ⋯
L(s)  = 1  + (1.03 − 1.29i)5-s + (0.365 + 0.930i)7-s + (−0.222 − 0.974i)9-s + (0.326 − 1.42i)11-s + (−1.48 + 0.716i)17-s + 19-s + (0.400 + 0.193i)23-s + (−0.385 − 1.68i)25-s + (1.57 + 0.487i)35-s + (0.0931 + 0.116i)43-s + (−1.48 − 0.716i)45-s + (0.440 − 1.92i)47-s + (−0.733 + 0.680i)49-s + (−1.51 − 1.89i)55-s + (−1.72 + 0.829i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.325 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.569261902\)
\(L(\frac12)\) \(\approx\) \(1.569261902\)
\(L(1)\) 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 \( 1 + (-0.365 - 0.930i)T \)
19 \( 1 - T \)
good3 \( 1 + (0.222 + 0.974i)T^{2} \)
5 \( 1 + (-1.03 + 1.29i)T + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.326 + 1.42i)T + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (1.48 - 0.716i)T + (0.623 - 0.781i)T^{2} \)
23 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.440 + 1.92i)T + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (1.72 - 0.829i)T + (0.623 - 0.781i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.222 - 0.974i)T + (-0.900 + 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
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

−8.675676446989356882626036840206, −8.269780975129914646654285149303, −6.87106974439490896458845826787, −6.02557010480850031375179488364, −5.69233992324915086752853042035, −4.96071731353891503337768340937, −3.97326905764013049878808233464, −2.94818279459110162499092144373, −1.88834638058049176666751927600, −0.932703206351005404216304251387, 1.65664633435774177368808196580, 2.35137677137856706087340377305, 3.17955674762974100122597815775, 4.50307209277044555878278696342, 4.88795390692779871869645698122, 6.01955599775134041719466119283, 6.76948698670214580206564754022, 7.31773564001676797163099574201, 7.75255922321978533610826242637, 9.084364424027943116258881557525

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