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Properties

Label 2-2100-35.17-c1-0-18
Degree $2$
Conductor $2100$
Sign $0.550 + 0.834i$
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 + (1.93 − 1.80i)7-s + (0.866 − 0.499i)9-s + (2.59 − 4.50i)11-s + (4.04 + 4.04i)13-s + (−0.327 − 1.22i)17-s + (−3.15 − 5.45i)19-s + (1.40 − 2.24i)21-s + (2.42 + 0.650i)23-s + (0.707 − 0.707i)27-s + 6.75i·29-s + (−8.26 − 4.77i)31-s + (1.34 − 5.02i)33-s + (1.55 − 5.80i)37-s + (4.95 + 2.86i)39-s + ⋯
L(s)  = 1  + (0.557 − 0.149i)3-s + (0.732 − 0.680i)7-s + (0.288 − 0.166i)9-s + (0.783 − 1.35i)11-s + (1.12 + 1.12i)13-s + (−0.0795 − 0.296i)17-s + (−0.722 − 1.25i)19-s + (0.307 − 0.488i)21-s + (0.506 + 0.135i)23-s + (0.136 − 0.136i)27-s + 1.25i·29-s + (−1.48 − 0.857i)31-s + (0.234 − 0.874i)33-s + (0.255 − 0.953i)37-s + (0.793 + 0.458i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.550 + 0.834i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.550 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.548281286\)
\(L(\frac12)\) \(\approx\) \(2.548281286\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 + 1.80i)T \)
good11 \( 1 + (-2.59 + 4.50i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.04 - 4.04i)T + 13iT^{2} \)
17 \( 1 + (0.327 + 1.22i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.15 + 5.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.42 - 0.650i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.75iT - 29T^{2} \)
31 \( 1 + (8.26 + 4.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.55 + 5.80i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 8.93iT - 41T^{2} \)
43 \( 1 + (2.26 - 2.26i)T - 43iT^{2} \)
47 \( 1 + (1.56 + 0.418i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.50 - 9.35i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.58 + 9.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.32 + 0.762i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.60 + 0.699i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + (-7.59 + 2.03i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (1.49 - 0.861i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.17 + 5.17i)T + 83iT^{2} \)
89 \( 1 + (-3.44 - 5.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.2 + 12.2i)T - 97iT^{2} \)
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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

−8.887818929736105154668232391551, −8.424080352982194543476721513720, −7.38713725006138507728643962976, −6.74503251523981285792840627410, −5.97027907752976569814596410069, −4.77264188508705049489457120874, −3.96892949656999466734591373590, −3.25372909430162650823908500163, −1.89571679250897013056127292615, −0.925722772950628792504477071348, 1.47620025038178601188151058536, 2.23224712207083156185129534571, 3.53732992510473699805804803487, 4.20984533505619332052928467556, 5.23893292000627301187130624356, 6.00398992040670102936817361029, 6.97936205656201750010803268015, 7.86657974320566825025119292077, 8.505428312994939675051345768661, 9.020100458097078365670174649723

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