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Properties

Label 2-603-67.40-c1-0-25
Degree $2$
Conductor $603$
Sign $-0.496 - 0.868i$
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  + (−0.898 − 1.03i)2-s + (0.0164 − 0.114i)4-s + (0.297 + 0.0872i)5-s + (−1.28 − 1.48i)7-s + (−2.44 + 1.57i)8-s + (−0.176 − 0.386i)10-s + (−1.41 − 0.414i)11-s + (−2.40 − 1.54i)13-s + (−0.384 + 2.67i)14-s + (3.60 + 1.05i)16-s + (0.675 + 4.69i)17-s + (−3.31 + 3.82i)19-s + (0.0148 − 0.0325i)20-s + (0.839 + 1.83i)22-s + (2.31 − 5.06i)23-s + ⋯
L(s)  = 1  + (−0.635 − 0.733i)2-s + (0.00821 − 0.0571i)4-s + (0.132 + 0.0390i)5-s + (−0.486 − 0.561i)7-s + (−0.863 + 0.555i)8-s + (−0.0558 − 0.122i)10-s + (−0.426 − 0.125i)11-s + (−0.667 − 0.428i)13-s + (−0.102 + 0.714i)14-s + (0.900 + 0.264i)16-s + (0.163 + 1.13i)17-s + (−0.761 + 0.878i)19-s + (0.00332 − 0.00727i)20-s + (0.179 + 0.392i)22-s + (0.481 − 1.05i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(603\)    =    \(3^{2} \cdot 67\)
Sign: $-0.496 - 0.868i$
Analytic conductor: \(4.81497\)
Root analytic conductor: \(2.19430\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{603} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 603,\ (\ :1/2),\ -0.496 - 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0607603 + 0.104739i\)
\(L(\frac12)\) \(\approx\) \(0.0607603 + 0.104739i\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.89 + 2.17i)T \)
good2 \( 1 + (0.898 + 1.03i)T + (-0.284 + 1.97i)T^{2} \)
5 \( 1 + (-0.297 - 0.0872i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (1.28 + 1.48i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (1.41 + 0.414i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (2.40 + 1.54i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-0.675 - 4.69i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (3.31 - 3.82i)T + (-2.70 - 18.8i)T^{2} \)
23 \( 1 + (-2.31 + 5.06i)T + (-15.0 - 17.3i)T^{2} \)
29 \( 1 + 7.60T + 29T^{2} \)
31 \( 1 + (1.07 - 0.691i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 - 0.658T + 37T^{2} \)
41 \( 1 + (1.14 + 7.96i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-1.74 - 12.1i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (4.10 - 8.98i)T + (-30.7 - 35.5i)T^{2} \)
53 \( 1 + (0.309 - 2.15i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (3.02 - 1.94i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-8.20 + 2.40i)T + (51.3 - 32.9i)T^{2} \)
71 \( 1 + (0.381 - 2.65i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (10.6 - 3.12i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (0.548 + 0.352i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (6.61 + 1.94i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (3.05 + 6.68i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 - 7.97T + 97T^{2} \)
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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

−10.30336635306387620984641156757, −9.465794707893944751391363098059, −8.470121848230472049153631181963, −7.64681316817422858355733856549, −6.34633596497501090489248378223, −5.62493646170770763354420765370, −4.18005710805549986231890510190, −2.93893343373837981583439905691, −1.75405296622268896890758592323, −0.07706590972374041151095123412, 2.36463525996506841411781129009, 3.52570416991534183306482559626, 5.05567256052955653586406983790, 5.97264951385665304439250739602, 7.10749630670431235276924280895, 7.48799997481468255552428037908, 8.731720115685836339648129130308, 9.333587548080490124739656325912, 9.923741221267681591814603269032, 11.33920862698662004095465613971

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