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Properties

Label 2-645-129.104-c1-0-20
Degree $2$
Conductor $645$
Sign $-0.800 - 0.599i$
Analytic cond. $5.15035$
Root an. cond. $2.26943$
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  + (2.24 + 1.08i)2-s + (−0.163 + 1.72i)3-s + (2.62 + 3.28i)4-s + (−0.733 + 0.680i)5-s + (−2.22 + 3.69i)6-s + (0.105 + 0.0606i)7-s + (1.22 + 5.34i)8-s + (−2.94 − 0.562i)9-s + (−2.37 + 0.734i)10-s + (0.790 + 0.630i)11-s + (−6.09 + 3.98i)12-s + (−1.86 − 0.576i)13-s + (0.170 + 0.249i)14-s + (−1.05 − 1.37i)15-s + (−1.17 + 5.12i)16-s + (2.37 − 2.56i)17-s + ⋯
L(s)  = 1  + (1.58 + 0.764i)2-s + (−0.0941 + 0.995i)3-s + (1.31 + 1.64i)4-s + (−0.327 + 0.304i)5-s + (−0.910 + 1.50i)6-s + (0.0396 + 0.0229i)7-s + (0.431 + 1.89i)8-s + (−0.982 − 0.187i)9-s + (−0.752 + 0.232i)10-s + (0.238 + 0.190i)11-s + (−1.75 + 1.14i)12-s + (−0.518 − 0.159i)13-s + (0.0454 + 0.0667i)14-s + (−0.271 − 0.355i)15-s + (−0.292 + 1.28i)16-s + (0.576 − 0.620i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(645\)    =    \(3 \cdot 5 \cdot 43\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(5.15035\)
Root analytic conductor: \(2.26943\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{645} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 645,\ (\ :1/2),\ -0.800 - 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01627 + 3.05496i\)
\(L(\frac12)\) \(\approx\) \(1.01627 + 3.05496i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.163 - 1.72i)T \)
5 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (-6.00 + 2.63i)T \)
good2 \( 1 + (-2.24 - 1.08i)T + (1.24 + 1.56i)T^{2} \)
7 \( 1 + (-0.105 - 0.0606i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.790 - 0.630i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (1.86 + 0.576i)T + (10.7 + 7.32i)T^{2} \)
17 \( 1 + (-2.37 + 2.56i)T + (-1.27 - 16.9i)T^{2} \)
19 \( 1 + (-4.25 - 1.66i)T + (13.9 + 12.9i)T^{2} \)
23 \( 1 + (0.405 - 2.69i)T + (-21.9 - 6.77i)T^{2} \)
29 \( 1 + (0.0809 - 0.0552i)T + (10.5 - 26.9i)T^{2} \)
31 \( 1 + (-0.110 - 1.47i)T + (-30.6 + 4.62i)T^{2} \)
37 \( 1 + (-2.89 + 1.67i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.856 - 1.77i)T + (-25.5 - 32.0i)T^{2} \)
47 \( 1 + (0.268 - 0.213i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-3.29 - 10.6i)T + (-43.7 + 29.8i)T^{2} \)
59 \( 1 + (2.50 + 0.571i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (9.04 + 0.677i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-0.301 + 0.768i)T + (-49.1 - 45.5i)T^{2} \)
71 \( 1 + (-1.39 + 0.210i)T + (67.8 - 20.9i)T^{2} \)
73 \( 1 + (-1.38 + 4.49i)T + (-60.3 - 41.1i)T^{2} \)
79 \( 1 + (-5.98 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.99 - 10.2i)T + (-30.3 - 77.2i)T^{2} \)
89 \( 1 + (9.56 + 6.51i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (-11.4 + 14.4i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.21911843499751041280887644170, −10.11497623301397892158417974694, −9.228570441586841100345421646885, −7.86937787959705598717858021097, −7.23508706109569764560405555639, −6.06919351541685308032896421706, −5.32484330442400341844489426517, −4.54219267322920884488910494722, −3.60168532799799229775297048253, −2.84062070055031258423267084252, 1.17869125483431668646931827405, 2.44894563954057119743261580109, 3.45938575095235633685603579592, 4.61312559732434479192406557071, 5.52725780278371072746418613912, 6.32692810867804958844165881255, 7.30881134902344709324646405261, 8.281028248569010948672401010974, 9.559979509553389148125466747322, 10.73452186086937891175108684618

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