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Properties

Label 2-3900-195.134-c0-0-3
Degree $2$
Conductor $3900$
Sign $-0.827 + 0.561i$
Analytic cond. $1.94635$
Root an. cond. $1.39511$
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  + (−0.866 − 0.5i)3-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.5i)13-s + 1.73i·21-s − 0.999i·27-s − 1.73i·31-s − 0.999·39-s + (0.866 − 0.5i)43-s + (−1 + 1.73i)49-s + (−0.5 − 0.866i)61-s + (0.866 − 1.49i)63-s + (−0.866 + 1.5i)67-s − 1.73·73-s − 79-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.5i)13-s + 1.73i·21-s − 0.999i·27-s − 1.73i·31-s − 0.999·39-s + (0.866 − 0.5i)43-s + (−1 + 1.73i)49-s + (−0.5 − 0.866i)61-s + (0.866 − 1.49i)63-s + (−0.866 + 1.5i)67-s − 1.73·73-s − 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.827 + 0.561i$
Analytic conductor: \(1.94635\)
Root analytic conductor: \(1.39511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (3449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :0),\ -0.827 + 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6540735710\)
\(L(\frac12)\) \(\approx\) \(0.6540735710\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-0.866 + 0.5i)T \)
good7 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.73T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.091575292840304659423139040058, −7.45620193140843911455963104681, −6.91339709220402566888259072572, −6.12318663383089238553139818382, −5.64819403591322058754033918367, −4.42101511386292128150086889307, −3.91301177406920720508049225338, −2.84392788195904368171993504686, −1.44411245643713356652101080664, −0.44596821867601550243976766429, 1.46246749003137747862370283856, 2.78382531189379408481489362641, 3.54298113151874833868796940180, 4.52171132470160609529831499496, 5.34655183821903547689086375623, 6.03170504697653979321636293058, 6.42867800199160574657923981366, 7.28222405317074638160418527363, 8.544585896602162959972208957014, 9.004648693020005029915723747406

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