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Properties

Label 2-605-55.2-c1-0-25
Degree $2$
Conductor $605$
Sign $-0.0551 - 0.998i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (1.15 + 2.25i)2-s + (0.313 − 1.98i)3-s + (−2.60 + 3.58i)4-s + (1.91 + 1.15i)5-s + (4.83 − 1.57i)6-s + (1.78 − 0.282i)7-s + (−6.07 − 0.962i)8-s + (−0.969 − 0.315i)9-s + (−0.412 + 5.65i)10-s + (6.27 + 6.27i)12-s + (2.00 − 1.02i)13-s + (2.69 + 3.70i)14-s + (2.89 − 3.42i)15-s + (−2.08 − 6.41i)16-s + (−2.99 − 1.52i)17-s + (−0.404 − 2.55i)18-s + ⋯
L(s)  = 1  + (0.813 + 1.59i)2-s + (0.181 − 1.14i)3-s + (−1.30 + 1.79i)4-s + (0.855 + 0.517i)5-s + (1.97 − 0.641i)6-s + (0.674 − 0.106i)7-s + (−2.14 − 0.340i)8-s + (−0.323 − 0.105i)9-s + (−0.130 + 1.78i)10-s + (1.81 + 1.81i)12-s + (0.557 − 0.284i)13-s + (0.720 + 0.991i)14-s + (0.746 − 0.884i)15-s + (−0.521 − 1.60i)16-s + (−0.726 − 0.370i)17-s + (−0.0953 − 0.601i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0551 - 0.998i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.0551 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84747 + 1.95239i\)
\(L(\frac12)\) \(\approx\) \(1.84747 + 1.95239i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.91 - 1.15i)T \)
11 \( 1 \)
good2 \( 1 + (-1.15 - 2.25i)T + (-1.17 + 1.61i)T^{2} \)
3 \( 1 + (-0.313 + 1.98i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 + (-1.78 + 0.282i)T + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (-2.00 + 1.02i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (2.99 + 1.52i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.05 + 0.767i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (3.16 - 3.16i)T - 23iT^{2} \)
29 \( 1 + (-2.37 - 1.72i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.23 - 3.81i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.501 - 3.16i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (0.766 + 1.05i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.55 + 4.55i)T + 43iT^{2} \)
47 \( 1 + (7.62 + 1.20i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (2.89 + 5.67i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-7.28 + 10.0i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-8.85 + 2.87i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (2.46 + 2.46i)T + 67iT^{2} \)
71 \( 1 + (2.09 + 6.43i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.47 - 9.32i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (0.353 - 1.08i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.75 + 9.32i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 - 3.85iT - 89T^{2} \)
97 \( 1 + (0.693 - 0.353i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.05395751065186025653889458938, −9.764529242184648576776985502762, −8.517086350674801700358656879465, −7.979370798207899711582217900756, −6.92208719643473841173014860220, −6.66328596041744022363237782205, −5.62015761553875338044821561546, −4.79413754656750468554172908382, −3.34057988266499641621045093843, −1.81123741678238821181849199324, 1.45546189187566517057041312698, 2.53318264068953293434617436852, 3.87907625768253531540413671469, 4.51336448367709182899583480410, 5.25538460796707190487511880766, 6.26703314417353199737354748358, 8.358773348572615419798013890010, 9.168990348389599410776562254869, 9.865653625179971328241513026321, 10.47443984677937059019427496230

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