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Properties

Label 2-605-11.9-c1-0-15
Degree $2$
Conductor $605$
Sign $-0.263 - 0.964i$
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  + (−0.780 + 2.40i)2-s + (1.99 − 1.44i)3-s + (−3.53 − 2.57i)4-s + (0.309 + 0.951i)5-s + (1.92 + 5.92i)6-s + (1.69 + 1.23i)7-s + (4.84 − 3.52i)8-s + (0.953 − 2.93i)9-s − 2.52·10-s − 10.7·12-s + (−0.398 + 1.22i)13-s + (−4.27 + 3.10i)14-s + (1.99 + 1.44i)15-s + (1.97 + 6.06i)16-s + (0.924 + 2.84i)17-s + (6.30 + 4.57i)18-s + ⋯
L(s)  = 1  + (−0.551 + 1.69i)2-s + (1.15 − 0.837i)3-s + (−1.76 − 1.28i)4-s + (0.138 + 0.425i)5-s + (0.785 + 2.41i)6-s + (0.640 + 0.465i)7-s + (1.71 − 1.24i)8-s + (0.317 − 0.977i)9-s − 0.798·10-s − 3.11·12-s + (−0.110 + 0.339i)13-s + (−1.14 + 0.830i)14-s + (0.515 + 0.374i)15-s + (0.492 + 1.51i)16-s + (0.224 + 0.689i)17-s + (1.48 + 1.07i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.263 - 0.964i$
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: \((2,\ 605,\ (\ :1/2),\ -0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.979714 + 1.28316i\)
\(L(\frac12)\) \(\approx\) \(0.979714 + 1.28316i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.780 - 2.40i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.99 + 1.44i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (-1.69 - 1.23i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.398 - 1.22i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.924 - 2.84i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.74 + 1.26i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.77T + 23T^{2} \)
29 \( 1 + (0.495 + 0.360i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.12 - 3.46i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.51 - 1.09i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.13 - 3.00i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.17T + 43T^{2} \)
47 \( 1 + (-5.91 + 4.29i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.909 - 2.79i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.25 - 3.81i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.155 - 0.477i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 7.80T + 67T^{2} \)
71 \( 1 + (3.49 + 10.7i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.99 + 6.53i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.65 - 5.09i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.35 + 10.3i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 4.32T + 89T^{2} \)
97 \( 1 + (0.108 - 0.334i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.57766227947041758106130532310, −9.429206669303333914920356041302, −8.731550309897278320662005838829, −8.246361262583764248586737095816, −7.28262907336628815469910995878, −6.90914600669780162454687268087, −5.77145642596932528855628467032, −4.77447414118880968746191014434, −3.05403658781030033015888649039, −1.55003809740655496993843740801, 1.14058286429948686107823912168, 2.52754858784706796778547670101, 3.37207712354381707996487462888, 4.29817863042374157458024112913, 5.14776705003834046561360568484, 7.41796772852907463802825306107, 8.313093021846371631653471384678, 8.988431098874572960749650590453, 9.602637675420963670779533061359, 10.28037695202517301851912135614

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