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Properties

Label 2-605-11.9-c1-0-5
Degree $2$
Conductor $605$
Sign $-0.998 - 0.0475i$
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  + (−0.127 + 0.393i)2-s + (−2.28 + 1.66i)3-s + (1.47 + 1.07i)4-s + (−0.309 − 0.951i)5-s + (−0.362 − 1.11i)6-s + (1.61 + 1.17i)7-s + (−1.28 + 0.932i)8-s + (1.54 − 4.75i)9-s + 0.414·10-s − 5.17·12-s + (−2.11 + 6.49i)13-s + (−0.670 + 0.486i)14-s + (2.28 + 1.66i)15-s + (0.927 + 2.85i)16-s + (0.362 + 1.11i)17-s + (1.67 + 1.21i)18-s + ⋯
L(s)  = 1  + (−0.0905 + 0.278i)2-s + (−1.32 + 0.959i)3-s + (0.739 + 0.537i)4-s + (−0.138 − 0.425i)5-s + (−0.147 − 0.454i)6-s + (0.611 + 0.444i)7-s + (−0.453 + 0.329i)8-s + (0.515 − 1.58i)9-s + 0.130·10-s − 1.49·12-s + (−0.585 + 1.80i)13-s + (−0.179 + 0.130i)14-s + (0.590 + 0.429i)15-s + (0.231 + 0.713i)16-s + (0.0878 + 0.270i)17-s + (0.394 + 0.286i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.998 - 0.0475i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.998 - 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0193206 + 0.812527i\)
\(L(\frac12)\) \(\approx\) \(0.0193206 + 0.812527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (0.127 - 0.393i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (2.28 - 1.66i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (-1.61 - 1.17i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (2.11 - 6.49i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.362 - 1.11i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + (6.19 + 4.50i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.95 + 2.14i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.85 - 3.52i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (-2.28 + 1.66i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.106 + 0.326i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.81 - 5.67i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.11 - 12.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4.48T + 67T^{2} \)
71 \( 1 + (3.49 + 10.7i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.52 - 4.01i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.23 + 3.80i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.85 + 5.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 9.31T + 89T^{2} \)
97 \( 1 + (2.36 - 7.28i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.36249375211899934476213357629, −10.36589074989020428835142670511, −9.349842206497225303421235810415, −8.572365113928528581736948429736, −7.35035970709838941731553247064, −6.50001342386788519154241195146, −5.56475153810326176742638549181, −4.71969345226234027610243291562, −3.81128150994161071035549951919, −1.97935923904923533723136566408, 0.53221621132894447870343386855, 1.74103430359740839961383725901, 3.14665280369323938819551441443, 5.13948882652340655720008387230, 5.59212786430599252718736167965, 6.75462189830479854739627057203, 7.25510981560922471869531164282, 8.078874247578821341463571942421, 9.805043027220898501064239104441, 10.68500144022398238735097799277

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