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Properties

Label 2-845-169.116-c1-0-15
Degree $2$
Conductor $845$
Sign $-0.747 - 0.664i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (0.663 + 0.749i)2-s + (0.696 + 1.83i)3-s + (0.120 − 0.991i)4-s + (0.239 + 0.970i)5-s + (−0.913 + 1.73i)6-s + (−2.27 + 1.56i)7-s + (2.46 − 1.70i)8-s + (−0.639 + 0.566i)9-s + (−0.568 + 0.823i)10-s + (−1.30 + 1.46i)11-s + (1.90 − 0.469i)12-s + (−3.05 + 1.91i)13-s + (−2.68 − 0.661i)14-s + (−1.61 + 1.11i)15-s + (0.977 + 0.240i)16-s + (2.32 + 3.37i)17-s + ⋯
L(s)  = 1  + (0.469 + 0.529i)2-s + (0.401 + 1.05i)3-s + (0.0601 − 0.495i)4-s + (0.107 + 0.434i)5-s + (−0.372 + 0.710i)6-s + (−0.858 + 0.592i)7-s + (0.873 − 0.602i)8-s + (−0.213 + 0.188i)9-s + (−0.179 + 0.260i)10-s + (−0.392 + 0.443i)11-s + (0.549 − 0.135i)12-s + (−0.846 + 0.532i)13-s + (−0.716 − 0.176i)14-s + (−0.417 + 0.287i)15-s + (0.244 + 0.0602i)16-s + (0.564 + 0.818i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.747 - 0.664i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711463 + 1.87127i\)
\(L(\frac12)\) \(\approx\) \(0.711463 + 1.87127i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.239 - 0.970i)T \)
13 \( 1 + (3.05 - 1.91i)T \)
good2 \( 1 + (-0.663 - 0.749i)T + (-0.241 + 1.98i)T^{2} \)
3 \( 1 + (-0.696 - 1.83i)T + (-2.24 + 1.98i)T^{2} \)
7 \( 1 + (2.27 - 1.56i)T + (2.48 - 6.54i)T^{2} \)
11 \( 1 + (1.30 - 1.46i)T + (-1.32 - 10.9i)T^{2} \)
17 \( 1 + (-2.32 - 3.37i)T + (-6.02 + 15.8i)T^{2} \)
19 \( 1 - 6.23iT - 19T^{2} \)
23 \( 1 - 0.920T + 23T^{2} \)
29 \( 1 + (6.16 - 5.46i)T + (3.49 - 28.7i)T^{2} \)
31 \( 1 + (-0.472 + 0.900i)T + (-17.6 - 25.5i)T^{2} \)
37 \( 1 + (-3.51 + 6.69i)T + (-21.0 - 30.4i)T^{2} \)
41 \( 1 + (-8.96 + 3.40i)T + (30.6 - 27.1i)T^{2} \)
43 \( 1 + (4.80 - 2.51i)T + (24.4 - 35.3i)T^{2} \)
47 \( 1 + (-12.2 + 1.48i)T + (45.6 - 11.2i)T^{2} \)
53 \( 1 + (4.99 + 7.24i)T + (-18.7 + 49.5i)T^{2} \)
59 \( 1 + (2.24 + 9.11i)T + (-52.2 + 27.4i)T^{2} \)
61 \( 1 + (2.55 - 3.69i)T + (-21.6 - 57.0i)T^{2} \)
67 \( 1 + (-10.2 + 1.24i)T + (65.0 - 16.0i)T^{2} \)
71 \( 1 + (3.98 - 1.51i)T + (53.1 - 47.0i)T^{2} \)
73 \( 1 + (-8.12 + 9.16i)T + (-8.79 - 72.4i)T^{2} \)
79 \( 1 + (-0.719 - 5.92i)T + (-76.7 + 18.9i)T^{2} \)
83 \( 1 + (-7.26 - 2.75i)T + (62.1 + 55.0i)T^{2} \)
89 \( 1 + 11.9iT - 89T^{2} \)
97 \( 1 + (1.66 - 6.77i)T + (-85.8 - 45.0i)T^{2} \)
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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

−10.31349795955767408912838219831, −9.644281112349501529524610984677, −9.248856498315642454398497942145, −7.81769121919491193191511629428, −6.93987565369943910612355295699, −5.98617735005751105748099539058, −5.31555858537626423364281370622, −4.21105050616116786885887549658, −3.43206246230904288619721956376, −2.05339253671332478624025854493, 0.793976176476845180304639341392, 2.46223439313181529010887029097, 3.00013247887334536326963191660, 4.30114743136830998094931507520, 5.28904946517202465153866097191, 6.61317115976002034640928382490, 7.53692295284386204792550747975, 7.79732561670035753552770365808, 9.001375658240420080170032815463, 9.889893159893521040041668529648

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