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Properties

Label 2-3100-155.154-c2-0-36
Degree $2$
Conductor $3100$
Sign $-0.247 - 0.968i$
Analytic cond. $84.4688$
Root an. cond. $9.19069$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.25·3-s + 6.21i·7-s − 7.43·9-s + 14.0i·11-s + 19.3·13-s + 28.3·17-s − 12.6·19-s + 7.77i·21-s + 5.27·23-s − 20.5·27-s + 34.3i·29-s + (23.4 − 20.2i)31-s + 17.5i·33-s + 34.6·37-s + 24.1·39-s + ⋯
L(s)  = 1  + 0.417·3-s + 0.888i·7-s − 0.825·9-s + 1.27i·11-s + 1.48·13-s + 1.66·17-s − 0.665·19-s + 0.370i·21-s + 0.229·23-s − 0.761·27-s + 1.18i·29-s + (0.755 − 0.654i)31-s + 0.532i·33-s + 0.935·37-s + 0.619·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $-0.247 - 0.968i$
Analytic conductor: \(84.4688\)
Root analytic conductor: \(9.19069\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{3100} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3100,\ (\ :1),\ -0.247 - 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.259925410\)
\(L(\frac12)\) \(\approx\) \(2.259925410\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
31 \( 1 + (-23.4 + 20.2i)T \)
good3 \( 1 - 1.25T + 9T^{2} \)
7 \( 1 - 6.21iT - 49T^{2} \)
11 \( 1 - 14.0iT - 121T^{2} \)
13 \( 1 - 19.3T + 169T^{2} \)
17 \( 1 - 28.3T + 289T^{2} \)
19 \( 1 + 12.6T + 361T^{2} \)
23 \( 1 - 5.27T + 529T^{2} \)
29 \( 1 - 34.3iT - 841T^{2} \)
37 \( 1 - 34.6T + 1.36e3T^{2} \)
41 \( 1 + 59.5T + 1.68e3T^{2} \)
43 \( 1 - 1.25T + 1.84e3T^{2} \)
47 \( 1 + 22.1iT - 2.20e3T^{2} \)
53 \( 1 + 31.5T + 2.80e3T^{2} \)
59 \( 1 - 41.9T + 3.48e3T^{2} \)
61 \( 1 - 46.8iT - 3.72e3T^{2} \)
67 \( 1 - 45.1iT - 4.48e3T^{2} \)
71 \( 1 - 67.2T + 5.04e3T^{2} \)
73 \( 1 + 48.6T + 5.32e3T^{2} \)
79 \( 1 + 151. iT - 6.24e3T^{2} \)
83 \( 1 + 87.7T + 6.88e3T^{2} \)
89 \( 1 - 98.0iT - 7.92e3T^{2} \)
97 \( 1 - 100. iT - 9.40e3T^{2} \)
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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

−8.617005621608535781889236217516, −8.218868013032844561430599695150, −7.34533212284051732672843374335, −6.37420179740905091697953891033, −5.72031571448213848876718267461, −5.00797396010465212765225799077, −3.88404059780571561110939352610, −3.10765682812244210504681708710, −2.23962906982412321080488698892, −1.22049720249565663410396347163, 0.51555149643290822178761592891, 1.36623993949154566827065520743, 2.86973091161757514196314818362, 3.46069810730142227106456295399, 4.14230807787655849522838774500, 5.41982752378758263606990359513, 6.02858958620665430901166397422, 6.70041639095550443774050828473, 7.932507909258023130474330630971, 8.197701325669318541069212902516

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