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Properties

Label 2-845-65.32-c1-0-63
Degree $2$
Conductor $845$
Sign $0.0663 - 0.997i$
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  + (0.915 − 1.58i)2-s + (−0.512 − 1.91i)3-s + (−0.677 − 1.17i)4-s + (−1.45 − 1.69i)5-s + (−3.50 − 0.939i)6-s + (−3.06 + 1.76i)7-s + 1.18·8-s + (−0.803 + 0.463i)9-s + (−4.02 + 0.752i)10-s + (−3.74 + 1.00i)11-s + (−1.89 + 1.89i)12-s + 6.48i·14-s + (−2.50 + 3.65i)15-s + (2.43 − 4.22i)16-s + (−1.95 − 0.524i)17-s + 1.69i·18-s + ⋯
L(s)  = 1  + (0.647 − 1.12i)2-s + (−0.296 − 1.10i)3-s + (−0.338 − 0.586i)4-s + (−0.650 − 0.759i)5-s + (−1.43 − 0.383i)6-s + (−1.15 + 0.668i)7-s + 0.417·8-s + (−0.267 + 0.154i)9-s + (−1.27 + 0.237i)10-s + (−1.12 + 0.302i)11-s + (−0.548 + 0.548i)12-s + 1.73i·14-s + (−0.646 + 0.943i)15-s + (0.609 − 1.05i)16-s + (−0.474 − 0.127i)17-s + 0.400i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0663 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0663 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.0663 - 0.997i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.0663 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.462031 + 0.432320i\)
\(L(\frac12)\) \(\approx\) \(0.462031 + 0.432320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.45 + 1.69i)T \)
13 \( 1 \)
good2 \( 1 + (-0.915 + 1.58i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.512 + 1.91i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (3.06 - 1.76i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.74 - 1.00i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.95 + 0.524i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.139 - 0.518i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.294 - 0.0788i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.71 + 0.988i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.13 - 4.13i)T - 31iT^{2} \)
37 \( 1 + (-4.69 - 2.70i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.174 - 0.649i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.28 + 8.51i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 9.75iT - 47T^{2} \)
53 \( 1 + (-3.16 + 3.16i)T - 53iT^{2} \)
59 \( 1 + (11.7 + 3.14i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.44 - 2.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.14 + 1.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.46 + 1.19i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 - 1.59iT - 79T^{2} \)
83 \( 1 + 7.57iT - 83T^{2} \)
89 \( 1 + (1.21 + 4.54i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.91 - 15.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.748592955307136884650929264593, −8.751980329439169596901423601252, −7.69735632012901605543510235927, −7.04461759710973368477403581856, −5.85671988096278435355945209524, −4.96868995911344192658396128086, −3.84422087202070512652490094679, −2.79593888563227527962995915612, −1.78213643688052248627814746204, −0.24591915397920695675431008948, 2.94870869159293006665707041836, 3.96631640765282247763558362768, 4.53762224242722962560017592762, 5.68875385839658355553194728574, 6.37478907779142291623911537846, 7.35731694974122093355858272758, 7.82179214061559175070534162395, 9.251079917322848794144731504122, 10.18547053611750609196626944123, 10.70679330992369766768969501467

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