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Properties

Label 2-4004-13.12-c1-0-26
Degree $2$
Conductor $4004$
Sign $0.251 - 0.967i$
Analytic cond. $31.9721$
Root an. cond. $5.65438$
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  + 1.49·3-s + 1.09i·5-s + i·7-s − 0.770·9-s + i·11-s + (0.907 − 3.48i)13-s + 1.63i·15-s − 2.55·17-s − 1.17i·19-s + 1.49i·21-s + 3.34·23-s + 3.80·25-s − 5.62·27-s + 3.03·29-s + 9.18i·31-s + ⋯
L(s)  = 1  + 0.862·3-s + 0.489i·5-s + 0.377i·7-s − 0.256·9-s + 0.301i·11-s + (0.251 − 0.967i)13-s + 0.421i·15-s − 0.619·17-s − 0.268i·19-s + 0.325i·21-s + 0.696·23-s + 0.760·25-s − 1.08·27-s + 0.562·29-s + 1.65i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.251 - 0.967i$
Analytic conductor: \(31.9721\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (2157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ 0.251 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.231925410\)
\(L(\frac12)\) \(\approx\) \(2.231925410\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - iT \)
11 \( 1 - iT \)
13 \( 1 + (-0.907 + 3.48i)T \)
good3 \( 1 - 1.49T + 3T^{2} \)
5 \( 1 - 1.09iT - 5T^{2} \)
17 \( 1 + 2.55T + 17T^{2} \)
19 \( 1 + 1.17iT - 19T^{2} \)
23 \( 1 - 3.34T + 23T^{2} \)
29 \( 1 - 3.03T + 29T^{2} \)
31 \( 1 - 9.18iT - 31T^{2} \)
37 \( 1 - 6.67iT - 37T^{2} \)
41 \( 1 - 3.39iT - 41T^{2} \)
43 \( 1 - 5.02T + 43T^{2} \)
47 \( 1 - 4.66iT - 47T^{2} \)
53 \( 1 - 9.15T + 53T^{2} \)
59 \( 1 + 1.21iT - 59T^{2} \)
61 \( 1 - 6.33T + 61T^{2} \)
67 \( 1 - 5.15iT - 67T^{2} \)
71 \( 1 - 7.39iT - 71T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 + 1.16T + 79T^{2} \)
83 \( 1 + 2.86iT - 83T^{2} \)
89 \( 1 - 3.44iT - 89T^{2} \)
97 \( 1 - 4.42iT - 97T^{2} \)
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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

−8.574832877027767808723086771089, −8.080910687130024807090019852905, −7.10732259912318120452901392568, −6.59016075103514636621452502672, −5.58087516042695947786543837165, −4.88791691387295821270665219090, −3.81421102830029527377776462096, −2.79336051983832464971565294394, −2.68956473811193907626058358141, −1.19756697992766458960919311181, 0.59741644244608566096159912931, 1.90914030128754818222551486260, 2.69894899409794953781217253079, 3.74620953179479373167675341906, 4.28990081553110533605431477290, 5.26867284931466873310025818583, 6.11840263807046411328698597703, 6.94215432222485717652565618121, 7.68214326394915942351086345573, 8.422375989073608927832813472293

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