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Properties

Label 1-712-712.595-r0-0-0
Degree $1$
Conductor $712$
Sign $-0.578 - 0.815i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (0.212 − 0.977i)3-s + (−0.755 + 0.654i)5-s + (0.997 + 0.0713i)7-s + (−0.909 − 0.415i)9-s + (0.654 − 0.755i)11-s + (−0.977 − 0.212i)13-s + (0.479 + 0.877i)15-s + (−0.281 − 0.959i)17-s + (−0.936 + 0.349i)19-s + (0.281 − 0.959i)21-s + (0.349 + 0.936i)23-s + (0.142 − 0.989i)25-s + (−0.599 + 0.800i)27-s + (0.0713 − 0.997i)29-s + (0.349 − 0.936i)31-s + ⋯
L(s)  = 1  + (0.212 − 0.977i)3-s + (−0.755 + 0.654i)5-s + (0.997 + 0.0713i)7-s + (−0.909 − 0.415i)9-s + (0.654 − 0.755i)11-s + (−0.977 − 0.212i)13-s + (0.479 + 0.877i)15-s + (−0.281 − 0.959i)17-s + (−0.936 + 0.349i)19-s + (0.281 − 0.959i)21-s + (0.349 + 0.936i)23-s + (0.142 − 0.989i)25-s + (−0.599 + 0.800i)27-s + (0.0713 − 0.997i)29-s + (0.349 − 0.936i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.578 - 0.815i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (595, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ -0.578 - 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4831875306 - 0.9356801882i\)
\(L(\frac12)\) \(\approx\) \(0.4831875306 - 0.9356801882i\)
\(L(1)\) \(\approx\) \(0.8846046976 - 0.3908765428i\)
\(L(1)\) \(\approx\) \(0.8846046976 - 0.3908765428i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.212 - 0.977i)T \)
5 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + (0.997 + 0.0713i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (-0.977 - 0.212i)T \)
17 \( 1 + (-0.281 - 0.959i)T \)
19 \( 1 + (-0.936 + 0.349i)T \)
23 \( 1 + (0.349 + 0.936i)T \)
29 \( 1 + (0.0713 - 0.997i)T \)
31 \( 1 + (0.349 - 0.936i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.977 + 0.212i)T \)
43 \( 1 + (-0.0713 - 0.997i)T \)
47 \( 1 + (0.540 - 0.841i)T \)
53 \( 1 + (-0.540 - 0.841i)T \)
59 \( 1 + (-0.212 - 0.977i)T \)
61 \( 1 + (-0.800 - 0.599i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (0.755 + 0.654i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (-0.909 + 0.415i)T \)
83 \( 1 + (0.479 - 0.877i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.88980460152244683915415301971, −21.83118176235492507656195016668, −21.334811968057965434850387570688, −20.31997133208021886384659637912, −19.90451146231955862159927452799, −19.136746201279124573261522245381, −17.66756423043009082902760413045, −17.05535930014990526235347689302, −16.42697290094393067983498957160, −15.258940773466361646773499033020, −14.86248527795164720073530073025, −14.181137888426149678763424225, −12.715457787421772358266718087664, −12.07983864540471272146160957172, −11.081368618189204381584714509142, −10.450755994875578437431360350929, −9.23765817215252256782164033695, −8.64482874693950855592498188359, −7.84728890915590731614863053727, −6.75581964007163597136168530707, −5.2828357698140719982237306126, −4.41400928478912101185480474872, −4.22126315475630885321119841466, −2.70618495619070922654582973465, −1.48145451864237648135291152184, 0.49253559252787973306611593025, 1.910836249032334170860515023516, 2.80623731280008672214628104240, 3.880509665466713307974947388308, 5.05667749925431999451592792168, 6.24112331928136801827345431034, 7.072644653320569567011732011976, 7.87917671406945258790651800222, 8.428000014143625961341840864639, 9.60259856504044863914506494950, 10.95639744676514796854085698392, 11.65059104503005478850391153331, 12.048151745357487315559970705543, 13.35674177228343469056906794859, 14.09131791160955208035979502720, 14.83057247340028345437331901440, 15.44128447727284496546658903235, 16.95666344434122328273099798808, 17.3865542402023965659372784025, 18.56201878346478375975763017509, 18.891590977019883611487744129975, 19.77725056280034424687389935164, 20.45073753285445641417155788010, 21.60838622146449397109057888027, 22.40787103287002025692083042457

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