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Properties

Label 2-2100-35.17-c1-0-22
Degree $2$
Conductor $2100$
Sign $-0.925 + 0.379i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)3-s + (−2.38 − 1.15i)7-s + (0.866 − 0.499i)9-s + (−1.36 + 2.36i)11-s + (0.707 + 0.707i)13-s + (1.74 + 6.50i)17-s + (−4.23 − 7.33i)19-s + (−2.59 − 0.500i)21-s + (−7.20 − 1.93i)23-s + (0.707 − 0.707i)27-s − 6.92i·29-s + (−1.73 − i)31-s + (−0.707 + 2.63i)33-s + (2.89 − 10.8i)37-s + (0.866 + 0.500i)39-s + ⋯
L(s)  = 1  + (0.557 − 0.149i)3-s + (−0.899 − 0.436i)7-s + (0.288 − 0.166i)9-s + (−0.411 + 0.713i)11-s + (0.196 + 0.196i)13-s + (0.422 + 1.57i)17-s + (−0.970 − 1.68i)19-s + (−0.566 − 0.109i)21-s + (−1.50 − 0.402i)23-s + (0.136 − 0.136i)27-s − 1.28i·29-s + (−0.311 − 0.179i)31-s + (−0.123 + 0.459i)33-s + (0.476 − 1.77i)37-s + (0.138 + 0.0800i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.925 + 0.379i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5252803234\)
\(L(\frac12)\) \(\approx\) \(0.5252803234\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (2.38 + 1.15i)T \)
good11 \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + 13iT^{2} \)
17 \( 1 + (-1.74 - 6.50i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (4.23 + 7.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.20 + 1.93i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + (1.73 + i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.89 + 10.8i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 4.19iT - 41T^{2} \)
43 \( 1 + (0.378 - 0.378i)T - 43iT^{2} \)
47 \( 1 + (9.84 + 2.63i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.26 - 8.43i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.63 + 8.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.42 - 4.86i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.76 - 1.81i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (14.8 - 3.98i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.96 - 1.13i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.34 - 3.34i)T + 83iT^{2} \)
89 \( 1 + (3.63 + 6.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.50 - 2.50i)T - 97iT^{2} \)
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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

−8.788857963324911411389834308804, −7.961610085252053879863307330880, −7.27747116457364452903101984490, −6.43352550284531970726450043765, −5.81573964566971062133843575486, −4.32124834549334559009826025832, −3.96831972380393004639721151420, −2.71559691468530690649417864914, −1.89362258429622780451814927028, −0.15801739954213586683814156182, 1.65304847822068120487576734936, 2.98190122573862929690182303810, 3.36596638557723930321306645490, 4.53594049355156865255520736859, 5.61824028046810492000003791057, 6.20060194271608370573966523719, 7.17273525289057169036036663493, 8.123738805772466513017732193575, 8.531109388594546586922901084278, 9.570465141843257012874823947405

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