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Properties

Label 2-1264-316.103-c1-0-26
Degree $2$
Conductor $1264$
Sign $0.498 + 0.866i$
Analytic cond. $10.0930$
Root an. cond. $3.17696$
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  + (0.214 − 0.370i)3-s + (0.323 + 0.561i)5-s + (−2.03 − 3.52i)7-s + (1.40 + 2.43i)9-s + (−0.217 + 0.125i)11-s + (−1.64 + 2.85i)13-s + 0.277·15-s − 1.82i·17-s + (5.79 − 3.34i)19-s − 1.74·21-s + (3.50 − 2.02i)23-s + (2.29 − 3.96i)25-s + 2.49·27-s + (3.46 − 2.00i)29-s + (−1.92 + 1.10i)31-s + ⋯
L(s)  = 1  + (0.123 − 0.214i)3-s + (0.144 + 0.250i)5-s + (−0.770 − 1.33i)7-s + (0.469 + 0.813i)9-s + (−0.0656 + 0.0379i)11-s + (−0.456 + 0.790i)13-s + 0.0716·15-s − 0.441i·17-s + (1.32 − 0.767i)19-s − 0.380·21-s + (0.731 − 0.422i)23-s + (0.458 − 0.793i)25-s + 0.479·27-s + (0.644 − 0.371i)29-s + (−0.344 + 0.199i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1264\)    =    \(2^{4} \cdot 79\)
Sign: $0.498 + 0.866i$
Analytic conductor: \(10.0930\)
Root analytic conductor: \(3.17696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1264} (735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1264,\ (\ :1/2),\ 0.498 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.586927278\)
\(L(\frac12)\) \(\approx\) \(1.586927278\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
79 \( 1 + (-5.88 + 6.66i)T \)
good3 \( 1 + (-0.214 + 0.370i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.323 - 0.561i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.03 + 3.52i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.217 - 0.125i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.64 - 2.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.82iT - 17T^{2} \)
19 \( 1 + (-5.79 + 3.34i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.50 + 2.02i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.46 + 2.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.92 - 1.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.78 + 1.02i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.9iT - 41T^{2} \)
43 \( 1 + (-1.51 + 2.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.32 + 2.28i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.27 - 2.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.25 + 7.37i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 0.126iT - 61T^{2} \)
67 \( 1 + 1.30iT - 67T^{2} \)
71 \( 1 + 8.95T + 71T^{2} \)
73 \( 1 + (-3.80 - 6.58i)T + (-36.5 + 63.2i)T^{2} \)
83 \( 1 + (1.06 - 0.612i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 1.93T + 97T^{2} \)
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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

−9.662161843661098275201090899624, −8.836798868760504892414614283643, −7.59911574800252258791262727296, −7.10118876187836806262059491502, −6.60349385534604359089609555532, −5.15746496656699613033703588843, −4.41646684720133498346588946182, −3.32333708430433292177304425636, −2.26648295932277482949464112808, −0.74244945034836664798775942308, 1.28666796503481677532339741664, 2.90307593572397710122096330484, 3.41505938750613945293039285905, 4.87376549852640225905657377274, 5.63712816644694191797240708860, 6.37135453283704564658803046391, 7.38019192264124594758327051640, 8.338157113095563268886429809073, 9.254126095450976199902694685957, 9.584201810568454233465974387237

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