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Properties

Label 2-605-55.14-c1-0-31
Degree $2$
Conductor $605$
Sign $0.596 + 0.802i$
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  + (−0.465 − 0.640i)2-s + (2.40 − 0.780i)3-s + (0.424 − 1.30i)4-s + (1.49 + 1.66i)5-s + (−1.61 − 1.17i)6-s + (3.29 + 1.07i)7-s + (−2.54 + 0.825i)8-s + (2.72 − 1.98i)9-s + (0.372 − 1.73i)10-s − 3.46i·12-s + (−0.848 − 2.61i)14-s + (4.88 + 2.83i)15-s + (−0.507 − 0.368i)16-s + (−2.96 + 4.08i)17-s + (−2.54 − 0.825i)18-s + (−1.23 − 3.80i)19-s + ⋯
L(s)  = 1  + (−0.329 − 0.453i)2-s + (1.38 − 0.450i)3-s + (0.212 − 0.652i)4-s + (0.667 + 0.744i)5-s + (−0.660 − 0.479i)6-s + (1.24 + 0.404i)7-s + (−0.898 + 0.291i)8-s + (0.909 − 0.660i)9-s + (0.117 − 0.547i)10-s − 0.999i·12-s + (−0.226 − 0.697i)14-s + (1.26 + 0.731i)15-s + (−0.126 − 0.0922i)16-s + (−0.719 + 0.990i)17-s + (−0.598 − 0.194i)18-s + (−0.283 − 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.596 + 0.802i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.596 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13927 - 1.07533i\)
\(L(\frac12)\) \(\approx\) \(2.13927 - 1.07533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.49 - 1.66i)T \)
11 \( 1 \)
good2 \( 1 + (0.465 + 0.640i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (-2.40 + 0.780i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-3.29 - 1.07i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.96 - 4.08i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.52iT - 23T^{2} \)
29 \( 1 + (-0.848 + 2.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.91 + 1.39i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (10.4 + 3.41i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.848 + 2.61i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (6.30 - 2.04i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.86 + 2.56i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.502 + 1.54i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.69 + 6.31i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.644iT - 67T^{2} \)
71 \( 1 + (5.75 + 4.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.58 - 2.14i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.3 + 7.49i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.89 + 5.36i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 + (-2.41 - 3.32i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.59169204927515295305594365162, −9.506978255358787689506415456427, −8.857779814486194281198954526850, −8.133140010129705150430913199908, −7.06550733485284871607264973473, −6.12803085540275702846176064858, −4.98104011223768014935510913513, −3.32761883065566446290228715362, −2.12158259263366230500902074374, −1.83571542942860539038903147852, 1.79526675794099255929946326871, 2.94739508367313241340350936265, 4.18411906471004461145124184817, 5.02697167023583249615528448042, 6.53725681409948575365286716014, 7.62561592366467383426422927133, 8.353745723102189645179353450170, 8.743685762351168563359875373025, 9.559437415749020636995977450147, 10.50804823373872136315432880077

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