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Properties

Label 2-603-67.14-c1-0-25
Degree $2$
Conductor $603$
Sign $-0.528 + 0.848i$
Analytic cond. $4.81497$
Root an. cond. $2.19430$
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  + (−0.0366 − 0.0801i)2-s + (1.30 − 1.50i)4-s + (−0.260 − 1.81i)5-s + (−0.253 − 0.554i)7-s + (−0.337 − 0.0991i)8-s + (−0.135 + 0.0872i)10-s + (−0.769 − 5.35i)11-s + (−5.48 + 1.61i)13-s + (−0.0351 + 0.0405i)14-s + (−0.562 − 3.91i)16-s + (3.76 + 4.34i)17-s + (−0.821 + 1.79i)19-s + (−3.06 − 1.97i)20-s + (−0.401 + 0.257i)22-s + (−2.95 − 1.90i)23-s + ⋯
L(s)  = 1  + (−0.0258 − 0.0567i)2-s + (0.652 − 0.752i)4-s + (−0.116 − 0.810i)5-s + (−0.0956 − 0.209i)7-s + (−0.119 − 0.0350i)8-s + (−0.0429 + 0.0275i)10-s + (−0.232 − 1.61i)11-s + (−1.52 + 0.446i)13-s + (−0.00940 + 0.0108i)14-s + (−0.140 − 0.978i)16-s + (0.912 + 1.05i)17-s + (−0.188 + 0.412i)19-s + (−0.686 − 0.440i)20-s + (−0.0854 + 0.0549i)22-s + (−0.617 − 0.396i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.528 + 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $-0.528 + 0.848i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ -0.528 + 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647962 - 1.16668i\)
\(L(\frac12)\) \(\approx\) \(0.647962 - 1.16668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
67 \( 1 + (-8.14 + 0.846i)T \)
good2 \( 1 + (0.0366 + 0.0801i)T + (-1.30 + 1.51i)T^{2} \)
5 \( 1 + (0.260 + 1.81i)T + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (0.253 + 0.554i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (0.769 + 5.35i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (5.48 - 1.61i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-3.76 - 4.34i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (0.821 - 1.79i)T + (-12.4 - 14.3i)T^{2} \)
23 \( 1 + (2.95 + 1.90i)T + (9.55 + 20.9i)T^{2} \)
29 \( 1 + 4.50T + 29T^{2} \)
31 \( 1 + (-3.92 - 1.15i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 - 3.02T + 37T^{2} \)
41 \( 1 + (-1.80 - 2.08i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (4.28 + 4.94i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (-1.86 - 1.19i)T + (19.5 + 42.7i)T^{2} \)
53 \( 1 + (-5.19 + 5.99i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (0.733 + 0.215i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-1.49 + 10.4i)T + (-58.5 - 17.1i)T^{2} \)
71 \( 1 + (-8.46 + 9.76i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (1.52 - 10.6i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (-11.3 + 3.34i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (0.467 + 3.25i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (4.42 - 2.84i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + 9.24T + 97T^{2} \)
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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

−10.32118719640612498286540493844, −9.676099999163613234576095183666, −8.560349244182230624765660695579, −7.81037087168369796434432607997, −6.63101023388425158866640344707, −5.75085989204730205493082648260, −4.97096034644315371443788539012, −3.59947838356394479538081397869, −2.18179639537375902277389449260, −0.71407386859022166056039411978, 2.32323364236448530636620553960, 2.93050252170428469120488699276, 4.34017921359205052724746282659, 5.48265003079740587086742268456, 6.88294328143989376744591304961, 7.32649570562257726371325907468, 7.925924005436937484993445657057, 9.443678372583873613656578043701, 10.02756477860655587063901130098, 10.98193360521002204388473123392

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