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Properties

Label 2-605-5.4-c1-0-32
Degree $2$
Conductor $605$
Sign $0.774 + 0.632i$
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  + 0.517i·2-s − 1.93i·3-s + 1.73·4-s + (1.73 + 1.41i)5-s + 0.999·6-s − 3.34i·7-s + 1.93i·8-s − 0.732·9-s + (−0.732 + 0.896i)10-s − 3.34i·12-s − 4.24i·13-s + 1.73·14-s + (2.73 − 3.34i)15-s + 2.46·16-s + 3.86i·17-s − 0.378i·18-s + ⋯
L(s)  = 1  + 0.366i·2-s − 1.11i·3-s + 0.866·4-s + (0.774 + 0.632i)5-s + 0.408·6-s − 1.26i·7-s + 0.683i·8-s − 0.244·9-s + (−0.231 + 0.283i)10-s − 0.965i·12-s − 1.17i·13-s + 0.462·14-s + (0.705 − 0.863i)15-s + 0.616·16-s + 0.937i·17-s − 0.0893i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96036 - 0.698662i\)
\(L(\frac12)\) \(\approx\) \(1.96036 - 0.698662i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.73 - 1.41i)T \)
11 \( 1 \)
good2 \( 1 - 0.517iT - 2T^{2} \)
3 \( 1 + 1.93iT - 3T^{2} \)
7 \( 1 + 3.34iT - 7T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 - 3.86iT - 17T^{2} \)
19 \( 1 + 4.19T + 19T^{2} \)
23 \( 1 - 3.48iT - 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 8.73T + 31T^{2} \)
37 \( 1 + 1.79iT - 37T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 - 6.45iT - 43T^{2} \)
47 \( 1 + 11.4iT - 47T^{2} \)
53 \( 1 - 2.17iT - 53T^{2} \)
59 \( 1 - 1.26T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + 2.20iT - 67T^{2} \)
71 \( 1 + 8.19T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 - 1.46T + 79T^{2} \)
83 \( 1 - 9.89iT - 83T^{2} \)
89 \( 1 + 0.464T + 89T^{2} \)
97 \( 1 - 9.14iT - 97T^{2} \)
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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

−10.53106711158502239059709071616, −10.06862280275070736369092029573, −8.375702864837947167056178882023, −7.59814086719099772641773812813, −6.95329211892590322979874332128, −6.36431477736149942927451140635, −5.47289900694261460548180292046, −3.67478777871946488444632706909, −2.37597067451157788861953431799, −1.32610852839714391020366130787, 1.84298477059044662802303793246, 2.76546245706646527449379079735, 4.22280645332290617148453434009, 5.12904671094793661632387214692, 6.08727604891512810215215301243, 6.96924107366894329723833464471, 8.593330131114498158373459493530, 9.171536057650515139715379666807, 9.875658047314959845240349284513, 10.66153376318042486512155941609

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