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Properties

Label 2-756-28.19-c1-0-5
Degree $2$
Conductor $756$
Sign $0.309 - 0.950i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (−0.271 − 1.38i)2-s + (−1.85 + 0.753i)4-s + (−0.945 − 0.545i)5-s + (2.64 + 0.133i)7-s + (1.54 + 2.36i)8-s + (−0.500 + 1.46i)10-s + (−4.97 + 2.86i)11-s − 1.80i·13-s + (−0.532 − 3.70i)14-s + (2.86 − 2.79i)16-s + (−4.79 + 2.76i)17-s + (−3.39 + 5.88i)19-s + (2.16 + 0.298i)20-s + (5.33 + 6.11i)22-s + (−1.99 − 1.14i)23-s + ⋯
L(s)  = 1  + (−0.192 − 0.981i)2-s + (−0.926 + 0.376i)4-s + (−0.422 − 0.244i)5-s + (0.998 + 0.0505i)7-s + (0.547 + 0.836i)8-s + (−0.158 + 0.461i)10-s + (−1.49 + 0.865i)11-s − 0.499i·13-s + (−0.142 − 0.989i)14-s + (0.715 − 0.698i)16-s + (−1.16 + 0.671i)17-s + (−0.779 + 1.34i)19-s + (0.483 + 0.0666i)20-s + (1.13 + 1.30i)22-s + (−0.415 − 0.239i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.309 - 0.950i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.309 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.379751 + 0.275848i\)
\(L(\frac12)\) \(\approx\) \(0.379751 + 0.275848i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.271 + 1.38i)T \)
3 \( 1 \)
7 \( 1 + (-2.64 - 0.133i)T \)
good5 \( 1 + (0.945 + 0.545i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.97 - 2.86i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.80iT - 13T^{2} \)
17 \( 1 + (4.79 - 2.76i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.39 - 5.88i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.99 + 1.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.26T + 29T^{2} \)
31 \( 1 + (-1.07 - 1.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.16 - 5.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.53iT - 41T^{2} \)
43 \( 1 - 7.52iT - 43T^{2} \)
47 \( 1 + (-1.77 + 3.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.57 - 11.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.56 + 2.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.69 - 3.86i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.51 + 0.872i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.88iT - 71T^{2} \)
73 \( 1 + (12.1 - 7.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.7 + 6.18i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 + (-0.249 - 0.143i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−10.32916131863142558915611784620, −10.17532242667226569378034816072, −8.549151283467024239619532283056, −8.271844161252501641444970216230, −7.47765585062126109139750637941, −5.83885412506107005214982521313, −4.69526336022359003406864481460, −4.22455360164347646052262912976, −2.67383724025020716917446408071, −1.69893147198143219584341433697, 0.24872755583193459323778738418, 2.36423244522205282318538075810, 4.01060706023390710810730465952, 4.92808284003298656419709085472, 5.66653770301517942410821294768, 6.94118426838175143372306162955, 7.47958517617959304959198399637, 8.501081258476558951013206470709, 8.852686953269595768472642828708, 10.16732014532749564177513940247

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