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Properties

Label 2-605-55.32-c1-0-23
Degree $2$
Conductor $605$
Sign $0.997 - 0.0721i$
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  + (0.848 − 0.848i)2-s + (−0.258 + 0.258i)3-s + 0.560i·4-s + (−0.152 − 2.23i)5-s + 0.438i·6-s + (−1.06 + 1.06i)7-s + (2.17 + 2.17i)8-s + 2.86i·9-s + (−2.02 − 1.76i)10-s + (−0.144 − 0.144i)12-s + (3.20 + 3.20i)13-s + 1.80i·14-s + (0.616 + 0.537i)15-s + 2.56·16-s + (2.50 − 2.50i)17-s + (2.43 + 2.43i)18-s + ⋯
L(s)  = 1  + (0.599 − 0.599i)2-s + (−0.149 + 0.149i)3-s + 0.280i·4-s + (−0.0680 − 0.997i)5-s + 0.179i·6-s + (−0.402 + 0.402i)7-s + (0.768 + 0.768i)8-s + 0.955i·9-s + (−0.639 − 0.557i)10-s + (−0.0418 − 0.0418i)12-s + (0.887 + 0.887i)13-s + 0.482i·14-s + (0.159 + 0.138i)15-s + 0.641·16-s + (0.608 − 0.608i)17-s + (0.573 + 0.573i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.997 - 0.0721i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.997 - 0.0721i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97055 + 0.0712057i\)
\(L(\frac12)\) \(\approx\) \(1.97055 + 0.0712057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.152 + 2.23i)T \)
11 \( 1 \)
good2 \( 1 + (-0.848 + 0.848i)T - 2iT^{2} \)
3 \( 1 + (0.258 - 0.258i)T - 3iT^{2} \)
7 \( 1 + (1.06 - 1.06i)T - 7iT^{2} \)
13 \( 1 + (-3.20 - 3.20i)T + 13iT^{2} \)
17 \( 1 + (-2.50 + 2.50i)T - 17iT^{2} \)
19 \( 1 - 7.43T + 19T^{2} \)
23 \( 1 + (-1.84 + 1.84i)T - 23iT^{2} \)
29 \( 1 - 0.772T + 29T^{2} \)
31 \( 1 + 6.48T + 31T^{2} \)
37 \( 1 + (2.92 + 2.92i)T + 37iT^{2} \)
41 \( 1 - 6.91iT - 41T^{2} \)
43 \( 1 + (-0.500 - 0.500i)T + 43iT^{2} \)
47 \( 1 + (-5.46 - 5.46i)T + 47iT^{2} \)
53 \( 1 + (-7.53 + 7.53i)T - 53iT^{2} \)
59 \( 1 + 12.2iT - 59T^{2} \)
61 \( 1 + 3.31iT - 61T^{2} \)
67 \( 1 + (3.34 + 3.34i)T + 67iT^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + (-6.73 - 6.73i)T + 73iT^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 + (2.80 + 2.80i)T + 83iT^{2} \)
89 \( 1 - 9.96iT - 89T^{2} \)
97 \( 1 + (8.11 + 8.11i)T + 97iT^{2} \)
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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

−11.04842352984266486206350388030, −9.773944649318948234730496664875, −8.986525594330108864322164641081, −8.085037845750534196420546576876, −7.23677978294154431934854363603, −5.65292432100323725708476495551, −5.00858767184522156086576841476, −4.05028486506037292566143744341, −2.96144809061977151835135497379, −1.57829449826187501371828759510, 1.09641437518840806746013668403, 3.28532444640534915742526986499, 3.84369124625331496287841352481, 5.59698780241680056673371854046, 5.92011658463420232837263329383, 7.11016180308962089895232534776, 7.38521801178593919229989009247, 8.926525993513062904246573580573, 10.07078391508301550366368036222, 10.45504378610091802692204150072

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