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Properties

Label 2-55e2-275.189-c0-0-0
Degree $2$
Conductor $3025$
Sign $-0.441 + 0.897i$
Analytic cond. $1.50967$
Root an. cond. $1.22868$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.11 + 0.363i)3-s + (0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.309 − 0.224i)9-s + (−0.690 + 0.951i)12-s + (0.690 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)20-s + (−1.11 − 1.53i)23-s + (−0.809 + 0.587i)25-s + (0.427 − 0.587i)27-s + 1.61·31-s + (0.118 − 0.363i)36-s + (−0.309 − 0.224i)45-s + 1.17i·48-s + (−0.309 − 0.951i)49-s + ⋯
L(s)  = 1  + (−1.11 + 0.363i)3-s + (0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.309 − 0.224i)9-s + (−0.690 + 0.951i)12-s + (0.690 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)20-s + (−1.11 − 1.53i)23-s + (−0.809 + 0.587i)25-s + (0.427 − 0.587i)27-s + 1.61·31-s + (0.118 − 0.363i)36-s + (−0.309 − 0.224i)45-s + 1.17i·48-s + (−0.309 − 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.441 + 0.897i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1564, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ -0.441 + 0.897i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7135598982\)
\(L(\frac12)\) \(\approx\) \(0.7135598982\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 - 1.61T + T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
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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

−8.568470895573945011775070189472, −7.979847040718645898727351354918, −6.94947577402007524482296430296, −6.12336195676097047098543122798, −5.76186571213258538116674209327, −4.74313443569703660565430130677, −4.40915099417236747819868023866, −2.92410799160123062764049241187, −1.73183334958585425876003177554, −0.49799628024251871968136391819, 1.53331026833581312341595983226, 2.71800794203768576559641806890, 3.46493852348180053160155429432, 4.45411191565769742309969227048, 5.69543461685589448378388155010, 6.23435294767713988268273033177, 6.76397338949564377161622903804, 7.60715420788440767045089583472, 7.952575832010147605768174614769, 9.145521923564931263638776342967

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