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Properties

Label 1-445-445.318-r0-0-0
Degree $1$
Conductor $445$
Sign $-0.138 + 0.990i$
Analytic cond. $2.06657$
Root an. cond. $2.06657$
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  + (0.989 + 0.142i)2-s + (−0.479 + 0.877i)3-s + (0.959 + 0.281i)4-s + (−0.599 + 0.800i)6-s + (0.349 + 0.936i)7-s + (0.909 + 0.415i)8-s + (−0.540 − 0.841i)9-s + (−0.415 − 0.909i)11-s + (−0.707 + 0.707i)12-s + (0.877 + 0.479i)13-s + (0.212 + 0.977i)14-s + (0.841 + 0.540i)16-s + (0.142 + 0.989i)17-s + (−0.415 − 0.909i)18-s + (−0.212 + 0.977i)19-s + ⋯
L(s)  = 1  + (0.989 + 0.142i)2-s + (−0.479 + 0.877i)3-s + (0.959 + 0.281i)4-s + (−0.599 + 0.800i)6-s + (0.349 + 0.936i)7-s + (0.909 + 0.415i)8-s + (−0.540 − 0.841i)9-s + (−0.415 − 0.909i)11-s + (−0.707 + 0.707i)12-s + (0.877 + 0.479i)13-s + (0.212 + 0.977i)14-s + (0.841 + 0.540i)16-s + (0.142 + 0.989i)17-s + (−0.415 − 0.909i)18-s + (−0.212 + 0.977i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(445\)    =    \(5 \cdot 89\)
Sign: $-0.138 + 0.990i$
Analytic conductor: \(2.06657\)
Root analytic conductor: \(2.06657\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{445} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 445,\ (0:\ ),\ -0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.442946480 + 1.659561927i\)
\(L(\frac12)\) \(\approx\) \(1.442946480 + 1.659561927i\)
\(L(1)\) \(\approx\) \(1.491738324 + 0.8276949096i\)
\(L(1)\) \(\approx\) \(1.491738324 + 0.8276949096i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.989 + 0.142i)T \)
3 \( 1 + (-0.479 + 0.877i)T \)
7 \( 1 + (0.349 + 0.936i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (0.877 + 0.479i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (-0.212 + 0.977i)T \)
23 \( 1 + (0.212 - 0.977i)T \)
29 \( 1 + (0.349 + 0.936i)T \)
31 \( 1 + (-0.977 + 0.212i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (-0.479 - 0.877i)T \)
43 \( 1 + (-0.936 - 0.349i)T \)
47 \( 1 + (0.959 + 0.281i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (0.877 - 0.479i)T \)
61 \( 1 + (-0.997 + 0.0713i)T \)
67 \( 1 + (0.281 + 0.959i)T \)
71 \( 1 + (0.909 - 0.415i)T \)
73 \( 1 + (0.540 - 0.841i)T \)
79 \( 1 + (0.540 - 0.841i)T \)
83 \( 1 + (0.800 + 0.599i)T \)
97 \( 1 + (-0.909 - 0.415i)T \)
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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

−23.48313145095747083661226656280, −23.20012247503010845320464567227, −22.395797462630310040664305822948, −21.2508225114156003889014883533, −20.3167941801548961277909706492, −19.83677464000522879995209484946, −18.625586162098644098754073279766, −17.718590881958430718466779429297, −16.91660307025040668399177468321, −15.857135079934343290312642785136, −14.98079291952184819556809059013, −13.62216213858425016066639159887, −13.51148000011742052156546437159, −12.48877338671964136181164556532, −11.46921672644585150790521025574, −10.93089198912233623358049409663, −9.8614004795860351395865742492, −8.03944914624121159380121089025, −7.25783741661177957633837709816, −6.57560725898834452912654832112, −5.3470897770781281216418889083, −4.66052779109355556186649852452, −3.32437569880437824708262451950, −2.105469787339060647666983483011, −1.01242607697533125376702912420, 1.79309142009750819552129726565, 3.21603816051909879434547344468, 3.95817637782476536432792314737, 5.15900899437282354837134802772, 5.79301641868507178937548913749, 6.55401420357235632946892850063, 8.23460279130487731703430676924, 8.89891967333559626537274807383, 10.58340417199136724195739895251, 10.92555616766295527150656524749, 12.0413744922572270325844825835, 12.66406930735157954403243754513, 14.0222522488680502317009937661, 14.697458977627720032216586699942, 15.56949367460722176061953665896, 16.26025978819979604612992180751, 16.92358149317475037932744788927, 18.21469360966860801922759987097, 19.13725341857959632850617939984, 20.5494184444097907599510295117, 21.08109870006863887717618635062, 21.77671954240469271901378386230, 22.33671589249792005586210858740, 23.505527324371846764999637517269, 23.874179590780307720169826774455

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