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Properties

Label 2-312-312.77-c0-0-3
Degree $2$
Conductor $312$
Sign $-0.707 + 0.707i$
Analytic cond. $0.155708$
Root an. cond. $0.394598$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s − i·3-s + 1.00i·4-s − 1.41i·5-s + (−0.707 + 0.707i)6-s + (0.707 − 0.707i)8-s − 9-s + (−1.00 + 1.00i)10-s + 1.41i·11-s + 1.00·12-s − i·13-s − 1.41·15-s − 1.00·16-s + (0.707 + 0.707i)18-s + 1.41·20-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s − i·3-s + 1.00i·4-s − 1.41i·5-s + (−0.707 + 0.707i)6-s + (0.707 − 0.707i)8-s − 9-s + (−1.00 + 1.00i)10-s + 1.41i·11-s + 1.00·12-s − i·13-s − 1.41·15-s − 1.00·16-s + (0.707 + 0.707i)18-s + 1.41·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(0.155708\)
Root analytic conductor: \(0.394598\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5398307506\)
\(L(\frac12)\) \(\approx\) \(0.5398307506\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + iT \)
13 \( 1 + iT \)
good5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.83184803506176836741157784678, −10.62433851693333639773189230935, −9.535566602470359230551740934978, −8.738181158848267996868063304094, −7.889187637506967363427835836916, −7.13928923669960564551078389804, −5.53767000079290569436689792216, −4.28171517576368719110760851354, −2.48585802223656674898890403744, −1.16159717848763700919032967287, 2.73814697579320246920588890563, 4.10959190140901163584825917625, 5.65890561338144717780429241510, 6.38836196830634882603924957532, 7.46504611769941035444126038159, 8.643214569290606881624117526876, 9.379941059344488175348511682378, 10.44601444652122138613385426467, 10.94423782161547508564549105562, 11.64158181560743975123646177508

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