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Properties

Label 2-312-104.99-c1-0-3
Degree $2$
Conductor $312$
Sign $0.899 - 0.437i$
Analytic cond. $2.49133$
Root an. cond. $1.57839$
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  + (0.437 − 1.34i)2-s − 3-s + (−1.61 − 1.17i)4-s + (−1.98 + 1.98i)5-s + (−0.437 + 1.34i)6-s + (3.05 + 3.05i)7-s + (−2.28 + 1.66i)8-s + 9-s + (1.79 + 3.52i)10-s + (−1.08 + 1.08i)11-s + (1.61 + 1.17i)12-s + (3.57 − 0.429i)13-s + (5.45 − 2.77i)14-s + (1.98 − 1.98i)15-s + (1.23 + 3.80i)16-s + 5.70i·17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s − 0.577·3-s + (−0.808 − 0.588i)4-s + (−0.885 + 0.885i)5-s + (−0.178 + 0.549i)6-s + (1.15 + 1.15i)7-s + (−0.809 + 0.587i)8-s + 0.333·9-s + (0.568 + 1.11i)10-s + (−0.327 + 0.327i)11-s + (0.466 + 0.339i)12-s + (0.992 − 0.119i)13-s + (1.45 − 0.742i)14-s + (0.511 − 0.511i)15-s + (0.308 + 0.951i)16-s + 1.38i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.899 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.947275 + 0.218387i\)
\(L(\frac12)\) \(\approx\) \(0.947275 + 0.218387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.437 + 1.34i)T \)
3 \( 1 + T \)
13 \( 1 + (-3.57 + 0.429i)T \)
good5 \( 1 + (1.98 - 1.98i)T - 5iT^{2} \)
7 \( 1 + (-3.05 - 3.05i)T + 7iT^{2} \)
11 \( 1 + (1.08 - 1.08i)T - 11iT^{2} \)
17 \( 1 - 5.70iT - 17T^{2} \)
19 \( 1 + (4.39 + 4.39i)T + 19iT^{2} \)
23 \( 1 - 2.95T + 23T^{2} \)
29 \( 1 - 6.96iT - 29T^{2} \)
31 \( 1 + (3.05 - 3.05i)T - 31iT^{2} \)
37 \( 1 + (-5.14 - 5.14i)T + 37iT^{2} \)
41 \( 1 + (2.77 + 2.77i)T + 41iT^{2} \)
43 \( 1 + 3.00iT - 43T^{2} \)
47 \( 1 + (6.40 + 6.40i)T + 47iT^{2} \)
53 \( 1 - 4.28iT - 53T^{2} \)
59 \( 1 + (-3.00 + 3.00i)T - 59iT^{2} \)
61 \( 1 + 1.13iT - 61T^{2} \)
67 \( 1 + (-7.39 - 7.39i)T + 67iT^{2} \)
71 \( 1 + (-3.20 + 3.20i)T - 71iT^{2} \)
73 \( 1 + (-7.87 + 7.87i)T - 73iT^{2} \)
79 \( 1 + 4.46iT - 79T^{2} \)
83 \( 1 + (9.78 + 9.78i)T + 83iT^{2} \)
89 \( 1 + (-4.24 + 4.24i)T - 89iT^{2} \)
97 \( 1 + (11.5 + 11.5i)T + 97iT^{2} \)
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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

−11.50726576110301797893353369416, −11.01336300915587088255967257715, −10.48800773803309989136065104331, −8.852512015768167393789431002497, −8.228809634982789571346025608021, −6.70889760928480621070753756501, −5.52969122964190126880825480064, −4.55617723034296507069085036807, −3.31264062656489494613761347156, −1.85363383783938000197977688079, 0.74052835578404071019817391054, 3.95518926832530471550891821479, 4.52482989425685172924443279059, 5.52193734311549541452936133266, 6.79419067163890073688045560687, 7.962566010611214540817822424761, 8.176973734317899665540432187120, 9.556903549825458244208542808138, 11.01038696078384983240478759719, 11.55942655154703544671924895254

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