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Properties

Label 2-644-7.4-c3-0-34
Degree $2$
Conductor $644$
Sign $-0.699 + 0.715i$
Analytic cond. $37.9972$
Root an. cond. $6.16418$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.38 + 5.86i)3-s + (−9.31 − 16.1i)5-s + (17.4 − 6.11i)7-s + (−9.46 − 16.3i)9-s + (12.9 − 22.4i)11-s − 12.5·13-s + 126.·15-s + (26.2 − 45.5i)17-s + (37.3 + 64.7i)19-s + (−23.3 + 123. i)21-s + (11.5 + 19.9i)23-s + (−111. + 192. i)25-s − 54.6·27-s − 164.·29-s + (71.5 − 123. i)31-s + ⋯
L(s)  = 1  + (−0.652 + 1.12i)3-s + (−0.833 − 1.44i)5-s + (0.943 − 0.330i)7-s + (−0.350 − 0.607i)9-s + (0.355 − 0.616i)11-s − 0.266·13-s + 2.17·15-s + (0.374 − 0.649i)17-s + (0.451 + 0.781i)19-s + (−0.242 + 1.28i)21-s + (0.104 + 0.180i)23-s + (−0.889 + 1.54i)25-s − 0.389·27-s − 1.05·29-s + (0.414 − 0.717i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.715i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.699 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.699 + 0.715i$
Analytic conductor: \(37.9972\)
Root analytic conductor: \(6.16418\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :3/2),\ -0.699 + 0.715i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5724864316\)
\(L(\frac12)\) \(\approx\) \(0.5724864316\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-17.4 + 6.11i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good3 \( 1 + (3.38 - 5.86i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (9.31 + 16.1i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-12.9 + 22.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 12.5T + 2.19e3T^{2} \)
17 \( 1 + (-26.2 + 45.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-37.3 - 64.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 + 164.T + 2.43e4T^{2} \)
31 \( 1 + (-71.5 + 123. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-18.5 - 32.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 277.T + 6.89e4T^{2} \)
43 \( 1 - 141.T + 7.95e4T^{2} \)
47 \( 1 + (110. + 191. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (232. - 403. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-142. + 246. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-77.1 - 133. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (0.345 - 0.598i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 658.T + 3.57e5T^{2} \)
73 \( 1 + (-433. + 751. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (41.7 + 72.2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 662.T + 5.71e5T^{2} \)
89 \( 1 + (569. + 986. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 242.T + 9.12e5T^{2} \)
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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

−9.810591436802828156220024160069, −9.058687347261390732722010213457, −8.151393872515162536631973759651, −7.47063697373523189904309073519, −5.72905573931638283335409790112, −5.07281168837406648420278297101, −4.37796143975357343866000562675, −3.61965482721705759533557415103, −1.37381155106389818272726966266, −0.19669351970762115637661559488, 1.40940598331952534073751082034, 2.56050732588996837606402804616, 3.86281388516671218715795351002, 5.15263335577564061466572285561, 6.28974245776722620297273267007, 7.06698207457963408031805231808, 7.52004395757275735865128082796, 8.402369861666356639706513416703, 9.787053719306571308355230084947, 10.88909143046055299693658598308

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