Properties

Label 2-6025-1.1-c1-0-1
Degree $2$
Conductor $6025$
Sign $1$
Analytic cond. $48.1098$
Root an. cond. $6.93612$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.895·2-s + 0.415·3-s − 1.19·4-s − 0.372·6-s − 2.78·7-s + 2.86·8-s − 2.82·9-s − 5.25·11-s − 0.498·12-s − 1.97·13-s + 2.49·14-s − 0.168·16-s − 6.98·17-s + 2.53·18-s + 8.09·19-s − 1.15·21-s + 4.70·22-s − 3.31·23-s + 1.19·24-s + 1.76·26-s − 2.42·27-s + 3.33·28-s + 1.04·29-s − 3.07·31-s − 5.57·32-s − 2.18·33-s + 6.25·34-s + ⋯
L(s)  = 1  − 0.633·2-s + 0.240·3-s − 0.599·4-s − 0.152·6-s − 1.05·7-s + 1.01·8-s − 0.942·9-s − 1.58·11-s − 0.143·12-s − 0.546·13-s + 0.666·14-s − 0.0420·16-s − 1.69·17-s + 0.596·18-s + 1.85·19-s − 0.252·21-s + 1.00·22-s − 0.690·23-s + 0.243·24-s + 0.346·26-s − 0.466·27-s + 0.630·28-s + 0.193·29-s − 0.553·31-s − 0.985·32-s − 0.380·33-s + 1.07·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $1$
Analytic conductor: \(48.1098\)
Root analytic conductor: \(6.93612\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.01458051689\)
\(L(\frac12)\) \(\approx\) \(0.01458051689\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 - T \)
good2 \( 1 + 0.895T + 2T^{2} \)
3 \( 1 - 0.415T + 3T^{2} \)
7 \( 1 + 2.78T + 7T^{2} \)
11 \( 1 + 5.25T + 11T^{2} \)
13 \( 1 + 1.97T + 13T^{2} \)
17 \( 1 + 6.98T + 17T^{2} \)
19 \( 1 - 8.09T + 19T^{2} \)
23 \( 1 + 3.31T + 23T^{2} \)
29 \( 1 - 1.04T + 29T^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
37 \( 1 + 5.38T + 37T^{2} \)
41 \( 1 + 6.04T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 - 5.22T + 47T^{2} \)
53 \( 1 + 4.91T + 53T^{2} \)
59 \( 1 + 5.43T + 59T^{2} \)
61 \( 1 + 9.07T + 61T^{2} \)
67 \( 1 + 8.20T + 67T^{2} \)
71 \( 1 - 5.40T + 71T^{2} \)
73 \( 1 + 8.17T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 9.14T + 97T^{2} \)
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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

−8.180102679030258740527083895571, −7.52589938143718325591038842957, −6.89657898404643544832586821926, −5.85924451323437723378101954282, −5.20815010384748925319216221993, −4.56936792584368426100990341658, −3.36300028567196030878590874969, −2.88465415216417112638280187378, −1.82838294505498075965953899879, −0.06509704169393321059836496555, 0.06509704169393321059836496555, 1.82838294505498075965953899879, 2.88465415216417112638280187378, 3.36300028567196030878590874969, 4.56936792584368426100990341658, 5.20815010384748925319216221993, 5.85924451323437723378101954282, 6.89657898404643544832586821926, 7.52589938143718325591038842957, 8.180102679030258740527083895571

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