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Properties

Label 2-845-65.28-c1-0-65
Degree $2$
Conductor $845$
Sign $-0.995 + 0.0996i$
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  + (1.80 − 1.04i)2-s + (0.713 − 2.66i)3-s + (1.17 − 2.04i)4-s + (−2.22 − 0.194i)5-s + (−1.48 − 5.55i)6-s + (1.45 − 2.52i)7-s − 0.750i·8-s + (−3.97 − 2.29i)9-s + (−4.23 + 1.97i)10-s + (−0.00681 + 0.0254i)11-s + (−4.59 − 4.59i)12-s − 6.07i·14-s + (−2.10 + 5.79i)15-s + (1.57 + 2.72i)16-s + (2.76 − 0.741i)17-s − 9.58·18-s + ⋯
L(s)  = 1  + (1.27 − 0.738i)2-s + (0.411 − 1.53i)3-s + (0.589 − 1.02i)4-s + (−0.996 − 0.0869i)5-s + (−0.607 − 2.26i)6-s + (0.550 − 0.952i)7-s − 0.265i·8-s + (−1.32 − 0.765i)9-s + (−1.33 + 0.624i)10-s + (−0.00205 + 0.00767i)11-s + (−1.32 − 1.32i)12-s − 1.62i·14-s + (−0.543 + 1.49i)15-s + (0.393 + 0.682i)16-s + (0.671 − 0.179i)17-s − 2.26·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.995 + 0.0996i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.995 + 0.0996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148181 - 2.96806i\)
\(L(\frac12)\) \(\approx\) \(0.148181 - 2.96806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.22 + 0.194i)T \)
13 \( 1 \)
good2 \( 1 + (-1.80 + 1.04i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.713 + 2.66i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.45 + 2.52i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.00681 - 0.0254i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.76 + 0.741i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.62 - 1.23i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.358 - 0.0961i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.62 + 2.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.835 + 0.835i)T - 31iT^{2} \)
37 \( 1 + (-3.22 - 5.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.57 + 2.02i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.79 + 6.69i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 0.833T + 47T^{2} \)
53 \( 1 + (-0.902 - 0.902i)T + 53iT^{2} \)
59 \( 1 + (0.387 + 1.44i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.35 + 9.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.6 + 6.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.957 + 3.57i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 15.0iT - 73T^{2} \)
79 \( 1 - 4.25iT - 79T^{2} \)
83 \( 1 + 1.31T + 83T^{2} \)
89 \( 1 + (3.23 + 0.867i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.351 - 0.202i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.18470758661346946929691766282, −8.395876665793104221387523960098, −8.088965221148522626493212234379, −7.14340885764464465147773975052, −6.38678083324100355674536080162, −5.07108083174844059704455061735, −4.13445575787222675665757773538, −3.28433751494848353387665093427, −2.13622824670605872763107958002, −0.977426736453809618221803231864, 2.79416101496199824779178197840, 3.63924022647383167302217473551, 4.46419115595297191079511465865, 4.99531706514008983387559648215, 5.89258422594714363278926331891, 7.02626210382514990640881318651, 8.194563550430690899110166920646, 8.671192846114977843700829906553, 9.759841628158983727805240508019, 10.66407259664774653049685333770

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