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Properties

Label 2-605-55.18-c1-0-35
Degree $2$
Conductor $605$
Sign $0.976 + 0.215i$
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  + (0.163 + 1.03i)2-s + (2.50 − 1.27i)3-s + (0.863 − 0.280i)4-s + (−1.84 − 1.27i)5-s + (1.72 + 2.38i)6-s + (0.249 − 0.489i)7-s + (1.37 + 2.70i)8-s + (2.89 − 3.98i)9-s + (1.01 − 2.10i)10-s + (1.80 − 1.80i)12-s + (3.20 − 0.507i)13-s + (0.546 + 0.177i)14-s + (−6.24 − 0.834i)15-s + (−1.09 + 0.798i)16-s + (−5.59 − 0.885i)17-s + (4.59 + 2.33i)18-s + ⋯
L(s)  = 1  + (0.115 + 0.729i)2-s + (1.44 − 0.738i)3-s + (0.431 − 0.140i)4-s + (−0.822 − 0.568i)5-s + (0.706 + 0.971i)6-s + (0.0943 − 0.185i)7-s + (0.487 + 0.957i)8-s + (0.966 − 1.32i)9-s + (0.319 − 0.666i)10-s + (0.522 − 0.522i)12-s + (0.888 − 0.140i)13-s + (0.146 + 0.0474i)14-s + (−1.61 − 0.215i)15-s + (−0.274 + 0.199i)16-s + (−1.35 − 0.214i)17-s + (1.08 + 0.551i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.976 + 0.215i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60257 - 0.283535i\)
\(L(\frac12)\) \(\approx\) \(2.60257 - 0.283535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.84 + 1.27i)T \)
11 \( 1 \)
good2 \( 1 + (-0.163 - 1.03i)T + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (-2.50 + 1.27i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-0.249 + 0.489i)T + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (-3.20 + 0.507i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (5.59 + 0.885i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-0.480 + 1.48i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.803 + 0.803i)T + 23iT^{2} \)
29 \( 1 + (-1.31 - 4.04i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.33 - 0.968i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.852 - 0.434i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (8.50 + 2.76i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (2.55 - 2.55i)T - 43iT^{2} \)
47 \( 1 + (-1.84 - 3.62i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (1.10 + 6.95i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-6.43 + 2.09i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.40 + 7.44i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.62 - 2.62i)T - 67iT^{2} \)
71 \( 1 + (5.54 - 4.02i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-8.98 - 4.58i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-3.37 - 2.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.77 - 11.1i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 - 3.64iT - 89T^{2} \)
97 \( 1 + (16.4 - 2.60i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.74572124067625552198850947599, −9.262815224846585394870348606198, −8.439055826334461444969064068963, −8.095455222641871359368943173785, −7.09560694103624171642404397136, −6.58288527977582179303944966924, −5.08409860753379918239005503718, −3.90931976268160511482130892040, −2.71609943515513823868212098263, −1.44577618949344002978168456881, 2.01810818156102110208563465169, 2.96811531934174265353171110286, 3.77799471901738438616919449116, 4.39716358917877895519240624267, 6.36476062890959942276936352145, 7.35895304183870302334120625353, 8.251993477805828289750989559166, 8.882972994246001822622326379290, 10.01668528486515242998727206239, 10.62697399714926159993896345769

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