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Properties

Label 2-2100-35.17-c1-0-1
Degree $2$
Conductor $2100$
Sign $-0.557 - 0.830i$
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 + (0.366 − 0.633i)11-s + (−0.707 − 0.707i)13-s + (−0.845 − 3.15i)17-s + (0.767 + 1.33i)19-s + (2.59 + 0.500i)21-s + (−0.517 − 0.138i)23-s + (−0.707 + 0.707i)27-s + 6.92i·29-s + (1.73 + i)31-s + (−0.189 + 0.707i)33-s + (0.208 − 0.776i)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 + (0.110 − 0.191i)11-s + (−0.196 − 0.196i)13-s + (−0.205 − 0.765i)17-s + (0.176 + 0.305i)19-s + (0.566 + 0.109i)21-s + (−0.107 − 0.0289i)23-s + (−0.136 + 0.136i)27-s + 1.28i·29-s + (0.311 + 0.179i)31-s + (−0.0329 + 0.123i)33-s + (0.0342 − 0.127i)37-s + (0.138 + 0.0800i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.557 - 0.830i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.557 - 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4373880917\)
\(L(\frac12)\) \(\approx\) \(0.4373880917\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.366 + 0.633i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.707 + 0.707i)T + 13iT^{2} \)
17 \( 1 + (0.845 + 3.15i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.767 - 1.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.517 + 0.138i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + (-1.73 - i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.208 + 0.776i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.19iT - 41T^{2} \)
43 \( 1 + (5.27 - 5.27i)T - 43iT^{2} \)
47 \( 1 + (0.189 + 0.0507i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.36 - 5.08i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.36 - 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.42 + 3.13i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.76 - 1.81i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (5.20 - 1.39i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.96 - 2.86i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.896 + 0.896i)T + 83iT^{2} \)
89 \( 1 + (-5.36 - 9.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.39 + 7.39i)T - 97iT^{2} \)
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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

−9.395306049706958289751899690849, −8.786789657092558710820512652042, −7.60858670006093100308370269170, −6.99256484990385155434360739209, −6.24599855072587096328731137104, −5.43910270495653758154983005364, −4.56343208851035122646684021528, −3.60864044835479833014016074296, −2.74816985693648907780965411537, −1.15442134288543521610766636677, 0.18200231802092872000545437765, 1.78189942044811172211619857868, 2.88056829542300328972157043928, 3.95147852703929537673393695020, 4.84651958315169360115991509869, 5.85246748716450222052464691987, 6.38519043577481475722851467782, 7.11920920790299941531227883754, 8.063249041680644013558871677938, 8.877452459149354840623734782319

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