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Properties

Label 2-845-65.18-c1-0-27
Degree $2$
Conductor $845$
Sign $0.100 + 0.994i$
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  − 2.64i·2-s + (0.917 + 0.917i)3-s − 5.02·4-s + (1.81 + 1.30i)5-s + (2.43 − 2.43i)6-s − 0.112·7-s + 8.00i·8-s − 1.31i·9-s + (3.45 − 4.81i)10-s + (1.31 + 1.31i)11-s + (−4.60 − 4.60i)12-s + 0.297i·14-s + (0.470 + 2.86i)15-s + 11.1·16-s + (1.93 + 1.93i)17-s − 3.49·18-s + ⋯
L(s)  = 1  − 1.87i·2-s + (0.529 + 0.529i)3-s − 2.51·4-s + (0.812 + 0.583i)5-s + (0.992 − 0.992i)6-s − 0.0424·7-s + 2.83i·8-s − 0.439i·9-s + (1.09 − 1.52i)10-s + (0.395 + 0.395i)11-s + (−1.32 − 1.32i)12-s + 0.0795i·14-s + (0.121 + 0.738i)15-s + 2.79·16-s + (0.468 + 0.468i)17-s − 0.823·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.100 + 0.994i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.100 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40767 - 1.27267i\)
\(L(\frac12)\) \(\approx\) \(1.40767 - 1.27267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.81 - 1.30i)T \)
13 \( 1 \)
good2 \( 1 + 2.64iT - 2T^{2} \)
3 \( 1 + (-0.917 - 0.917i)T + 3iT^{2} \)
7 \( 1 + 0.112T + 7T^{2} \)
11 \( 1 + (-1.31 - 1.31i)T + 11iT^{2} \)
17 \( 1 + (-1.93 - 1.93i)T + 17iT^{2} \)
19 \( 1 + (-4.92 - 4.92i)T + 19iT^{2} \)
23 \( 1 + (-2.27 + 2.27i)T - 23iT^{2} \)
29 \( 1 + 4.65iT - 29T^{2} \)
31 \( 1 + (0.624 - 0.624i)T - 31iT^{2} \)
37 \( 1 - 1.47T + 37T^{2} \)
41 \( 1 + (-3.83 + 3.83i)T - 41iT^{2} \)
43 \( 1 + (-2.75 + 2.75i)T - 43iT^{2} \)
47 \( 1 + 0.345T + 47T^{2} \)
53 \( 1 + (3.59 + 3.59i)T + 53iT^{2} \)
59 \( 1 + (-0.908 + 0.908i)T - 59iT^{2} \)
61 \( 1 + 2.78T + 61T^{2} \)
67 \( 1 - 0.144iT - 67T^{2} \)
71 \( 1 + (3.87 - 3.87i)T - 71iT^{2} \)
73 \( 1 - 9.06iT - 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 - 8.53T + 83T^{2} \)
89 \( 1 + (0.402 - 0.402i)T - 89iT^{2} \)
97 \( 1 - 14.9iT - 97T^{2} \)
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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

−10.04209503178728010669298221914, −9.519915592723884840423396470805, −8.934130480330269149249678132956, −7.80518873600507897631304647259, −6.30398843321332726888437834164, −5.20819704107895535377270527129, −4.00837158733227568238689047160, −3.36237050987210521723279981454, −2.45055657953710593929773694665, −1.29587105592671260920213878213, 1.15111953299883969595790158844, 3.04834580092530890253456249238, 4.68370469536453810147310553907, 5.27809243068815337809343024658, 6.14466812683148506030205797911, 7.07407461431135785998431397190, 7.67710797271602579222238744632, 8.536396548774237283709304501856, 9.213972099436833036143886851231, 9.720772786800470265084013416749

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